# Leveraging Speaker Embeddings in End-to-End Neural Diarization for Two-
Speaker Scenarios

###### Abstract

End-to-end neural speaker diarization systems are able to address the speaker
diarization task while effectively handling speech overlap. This work explores
the incorporation of speaker information embeddings into the end-to-end
systems to enhance the speaker discriminative capabilities, while maintaining
their overlap handling strengths. To achieve this, we propose several methods
for incorporating these embeddings along the acoustic features. Furthermore,
we delve into an analysis of the correct handling of silence frames, the
window length for extracting speaker embeddings and the transformer encoder
size. The effectiveness of our proposed approach is thoroughly evaluated on
the CallHome dataset for the two-speaker diarization task, with results that
demonstrate a significant reduction in diarization error rates achieving a
relative improvement of a 10.78% compared to the baseline end-to-end model.

##  1 Introduction

Speaker diarization, a crucial task in multiple speakers scenarios, involves
identifying different speaker segments within audio recordings [1].
Traditional approaches typically segment audio into speech and non-speech
chunks using Voice Activity Detection (VAD) [2] before extracting speaker
features from each speech segment. Utilizing VAD is essential as speaker
embedding extractors are not typically trained to effectively represent
silence. Among the commonly used speaker representations are i-vectors [3, 4],
d-vectors [5], x-vectors [6] and ECAPA-TDNN x-vectors [7], which encapsulate
information about speaker identity, speech style, and other related attributes
[8].

In the traditional modular diarization systems, these speaker embeddings are
then clustered using conventional clustering algorithm such as K-means [9],
Agglomerative Hierarchical Clustering [4, 6], spectral clustering [10] and
Bayesian HMM clustering of x-vector sequences (VBx) [11]. However, traditional
diarization strategies often struggle with overlapping speech, where multiple
speakers talking simultaneously pose a challenge to isolate and cluster
individual speaker information. Also, these speaker representations are often
extracted using a fixed-size sliding window. The window size can significantly
affect the diarization performance: while using larger window sizes helps
capture more speaker information, it increases the likelihood of including
multiple speakers, complicating clustering approaches. In contrast, smaller
window sizes reduce this risk, but may not provide enough data to extract
reliable speaker information. This trade-off between capturing speaker-
specific details and minimizing overlap-induced errors underscores the
importance of carefully selecting window sizes in diarization systems [12].

To address the challenge of speaker overlap, recent advancements in speaker
diarization have introduced end-to-end models, treating the task as a multi-
label classification problem [13, 14]. Models like the self-attentive end-to-
end neural diarization (SA-EEND) [15, 16] integrate various aspects like voice
activity and overlap detection, showing promising results at handling these
issues natively. However, these models encounter difficulties with scenarios
involving varying numbers of speakers. To overcome this limitation, newer
models like end-to-end neural diarization with encoder-decoder attractors
(EEND-EDA) [17, 18] have been developed to adapt dynamically to a variable
speaker number. Additionally, hybrid systems combining end-to-end approaches
with clustering algorithms have emerged achieving improved performance across
diverse scenarios [19, 20, 21, 22, 23]. Similarly, in [24, 25] speaker turns
are detected at word level by an automatic speech recognition (ASR) system to
subsequently perform speaker clustering diarization.

Another method to tackle the diarization task is to condition voice activity
detection with speaker embedding information. This approach leverages speaker
embeddings like i-vectors or x-vectors, concatenating them with acoustic
features of speech segments and feeding them into the model. Systems such as
Target-Speaker Voice Activity Detection (TS-VAD) [26] utilize this technique.
Additionally, methods described in [27, 28] demonstrate the effectiveness of
jointly training EEND and TS-VAD components.

In this study, inspired by the capabilities of TS-VAD, we aim to enhance the
speaker discriminative potential of the EEND-EDA diarization system by
incorporating speaker information extracted from a pre-trained ECAPA-TDNN [7]
x-vector extractor.

We have explored different ways to integrate the speaker embeddings at various
points of the architecture, aiming to help the model to better leverage the
speaker information. Additionally, we have focused on critical speaker
embedding extraction hyperparameters such as the window size, for optimal
performance in this task. Furthermore, to address the potential limitations of
speaker embeddings in representing silent segments, we applied an oracle VAD
to the speaker embeddings during the diarization training process, ensuring a
better handling of silent segments within the model. Finally, we used an
external VAD during testing for a comprehensive evaluation independent of
oracle VAD labels.

The rest of this paper is structured as follows: Section 2 explains the
speaker diarization pipeline, detailing the architecture and training scheme
of the proposed systems. Section 3 presents the experiments conducted and
their corresponding results are shown in Section 4. Finally, the paper
concludes with a summary in Section 5.

##  2 Methods

###  2.1 Review of End-to-End Neural Diarization with Encoder-Decoder
Attractors

Here we provide a concise overview of the end-to-end diarization framework
known as EEND-EDA [17]. The EEND module processes input data comprising a
sequence of log-scaled Mel-Filterbank features
$X=[x_{1},x_{2},...,x_{T}]\in\mathbb{R}^{T\times F}$, where $T$ is the
sequence length and $F$ the feature dimension. These features are passed
through either bi-directional long short-term memory (BLSTM) [13], Transformer
[15, 16], or Conformer [29] encoders to obtain embeddings
$e=[e_{1},e_{2},...,e_{T}]\in\mathbb{R}^{T\times D}$ at each time step.

To determine a flexible number of attractor points from variable-length
embedding sequences, an LSTM-based encoder-decoder architecture is employed.
The embedding sequence $(e_{t})^{T}_{t=1}$, with $D$ dimensions, is fed into
the unidirectional LSTM encoder.

$h_{0},c_{0}=LSTM_{encoder}(e_{1},...,e_{T})$ (1)

Attractors are generated during the decoding process.

$h_{s},c_{s},a_{s}=LSTM_{decoder}(h_{s-1},c_{s-1},0)$ (2)

These attractors are defined as $A=[a_{1},...,a_{s}]\in\mathbb{R}^{D\times S}$
where $S$ is the number of speakers. The probability of the presence of an
attractor $a$ is determined using a fully-connected layer with a sigmoid
activation function, defined as:

$p_{s}=\dfrac{1}{1+exp(-(w^{T}a_{s}+b))}$ (3)

Here, $w$ and $b$ represent the trainable weights and bias of the fully-
connected layer, respectively. During training, ground truth labels are
defined based on the actual number of speakers $S$.

$l_{s}=\begin{cases}1&\text{if }s\in\\{1,\ldots,S\\}\\\ 0&\text{if
}s=S+1\end{cases}$ (4)

The attractor existence loss $L_{\alpha}$ is calculated between $\hat{y}$ and
the ground truth labels $y$ using the binary cross entropy defined as:

$H(y_{t},\hat{y}_{t}):=\sum_{s}\left(-y_{t,s}\log\hat{y}_{t,s}-(1-y_{t,s})\log(1-\hat{y}_{t,s})\right)$
(5)

Therefore:

$L_{\alpha}=\dfrac{1}{1+S}H(l,p)$ (6)

Finally, the diarization loss is computed using the Permutation Invariant
Training (PIT) scheme [30]. This loss is calculated between $\hat{y}_{t}$ and
the ground truth labels $y_{t}=[y_{t,1}...,x_{t,S}]^{T}\in\\{0,1\\}^{S}$ as:

$L_{d}=\frac{1}{TS}\operatorname*{arg\,min}_{\phi\in\text{perm}(1,...,S)}\sum_{t=1}^{T}H(y_{t}^{\phi},\hat{y}_{t})$
(7)

The final loss is defined by the diarization loss and the attractor existence
loss.

$L=L_{d}+\alpha L_{\alpha}$ (8)

Where $\alpha$ is the weighting parameter for the attractor loss and it has a
value of $1$ for training and $0.1$ for adaptation.

###  2.2 ECAPA-TDNN speaker embedding extractor

The ECAPA-TDNN architecture [7] builds upon the well-established x-vector
topology [6] and integrates several enhancements to produce more robust
speaker embeddings. During training, the network is optimized to directly
minimize the cosine distance between speaker embeddings by optimizing the AAM-
softmax loss [31]. This approach makes it less dependent on complex scoring
backends such as Probabilistic Linear Discriminant Analysis (PLDA) [32]. The
ECAPA-TDNN model is trained using data from VoxCeleb1 and VoxCeleb2 [33, 34],
with additional data augmentation from the RIRs 111https://www.openslr.org/28/
and MUSAN [35] datasets, further enhancing its performance and generalization
capabilities.

Figure 1: Architecture of the EEND-EDA model with our different proposed
methods to integrate speaker embeddings into the system: a) Speaker embeddings
into EDA module described in section 2.3.1, b) Speaker embeddings into SA-EEND
encoder described in section 2.3.2, and c) Concatenation of speaker embeddings
and MFbank into SA-EEND encoder as described in section 2.3.3.

To ensure alignment between acoustic features and speaker embeddings, we
construct two sequences with the same number of frames. The acoustic feature
sequence is $X=[x_{1},x_{2},...,x_{T}]\in\mathbb{R}^{T\times F}$, and the
corresponding speaker embedding sequence is
$B=[b_{1},b_{2},...,b_{T}]\in\mathbb{R}^{T\times E}$. We use the same acoustic
features as in [17], and we adjust the sliding window of the ECAPA-TDNN
embedding extractor to match the number of frames of the acoustic feature
sequence.

###  2.3 Proposed methods

We propose three methods to combine the acoustic features $X$ and the speaker
embedding $B$ sequences as input to the SA-EEND-EDA model. In Figure 1, the
three proposed methods are depicted, distinguishable by the color of the
arrows (i.e., red arrows indicate method a), where Mel-Filterbank (MFbank)
goes to the SA-EEND module and the speaker embeddings to the EDA module).

####  2.3.1 Speaker embeddings into EDA module

As previously discussed in Section 2.1, the original encoder-decoder attractor
(EDA) module receives as input the output vector sequence from the last SA-
EEND transformer encoder block. This vector sequence, is then used for both
the diarization task (optimized using the PIT loss) and the speaker
identification or existence task within the EDA module. Specifically, the EDA
aims to identify the active speakers within the corresponding frame segment.
Once the attractors are calculated, they are used to condition the computation
of the diarization posteriors applying the dot-product operation with the same
encoder output vector sequence $e$. Our objective is to examine the behavior
of the model by directly introducing the speaker embeddings $B$ calculated
with the ECAPA-TDNN extractor into the EDA module as shown in Figure 1 a).
These embedding sequence $B$ are encoded with a single transformer encoder
block to maintain the original input dimension of the EDA module, ensuring
that:

$e_{spk}=\textit{Emb Encoder}(B)\in\mathbb{R}^{T\times D}$ (9)

In this setup, the SA-EEND transformer encoder takes just the Mel-Filterbank
acoustic feature sequence $X$ as input. Thus, we aim to determine if including
the speaker embedding information in the EDA module helps to better determine
the attractors and represent speaker variability.

####  2.3.2 Speaker embeddings into SA-EEND encoder

We also explore how effective speaker embeddings $B$ are on their own as input
to the end-to-end speaker diarization system, without using any acoustic
features, and if the model is able to refine the speaker discriminative
information within these embeddings for the task of diarization. To do this,
we use these ECAPA-TDNN embeddings $B$ as input features, extracted at the
same time resolution as $X$. In this setup, as in the SA-EEND-EDA baseline
[17], encoder output vector sequence $e$ are directly fed into the EDA module
as shown in Figure 1 b).

####  2.3.3 Concatenation of speaker embeddings and MFbank into SA-EEND
encoder

Taking inspiration from the TS-VAD model’s approach of combining speaker
embeddings with acoustic features, our setup aims to work on this idea by
concatenating the ECAPA-TDNN embedding sequence $B$ with Mel-filterbank
acoustic feature sequence $X$, and using it as input to the EEND-EDA model.
This concatenated sequence is denoted as:

$C=X\oplus B\in\mathbb{R}^{T\times(F+E)}$ (10)

Where $\oplus$ refers to the concatenation operation. With this configuration,
the model can utilize both feature sequences simultaneously, leading to an
integration of acoustic and speaker-related information. This setup is
illustrated in Figure 1 c).

###  2.4 Silence frames handling

Speaker embeddings are not typically trained to effectively discriminate
silence [36]. Extracting these embeddings directly with a sliding window
without applying VAD introduces the silence variability into the speaker
embedding sequence, complicating the subsequent speaker modeling within the
diarization system. Hence, to train the three proposed methods, we use the
oracle VAD segmentation to the speaker embeddings beforehand, replacing the
embeddings that correspond to silent segments by a zero vector
$\overrightarrow{0}$ of the same dimension. We also examined the scenario
where this oracle VAD segmentation was not used during training to assess
whether the model could learn to discard the silence segments without needing
the information of the VAD.

Additionally, we evaluated the models in CallHome without using VAD, using the
oracle VAD segmentation, and with an external VAD. As external VAD, we used
Kaldi’s [37] energy-based VAD.

##  3 Experimental setup

###  3.1 Data

For training the models we generated the sim2spk (simulated conversations with
2 speakers) dataset, using the generation algorithm outlined in [38], choosing
an speech overlap ratio of 34.4%. Specifically, sim2spk was constructed from
recordings extracted from Switchboard-2 (Phases I, II, and III), Switchboard
Cellular (Part1 and Part2) [39], and NIST Speaker Recognition Evaluation
(2004, 2005, 2006, and 2008) corpora [40, 41, 42, 43], all sampled at 8 kHz.
As in [16], each utterance was augmented with background noise from the MUSAN
dataset [35], as well as having a 50% probability of being convoluted with a
randomly chosen simulated room impulse response from the
RIR222https://www.openslr.org/28/ dataset [44]. To adapt the models trained on
simulated data to real conversations, we reserved the “Adapt” subset of the
2-speaker CallHome CH1-2spk telephone conversation dataset [45]. Finally, for
evaluation purposes, two distinct datasets were created: a simulated one,
generated with the same process as described before; and the remaining ”Test”
CH2-2spk subset of the 2-speakers CallHome dataset.

The details of the different datasets for training, testing and adaptation are
shown in Table 1.

Datasets | Sim2spk | CallHome
---|---|---
Train | Test | Adapt | Test
Total audio (h) | 2480 | 12.4 | 3.19 | 2.97
Average overlap (%) | 34.4 | 34.4 | 14 | 13.1
Table 1: Datasets used for training, adaptation and evaluation.

###  3.2 Experimental settings

The acoustic features extracted are 23-dimensional log-Mel-Filterbanks with a
frame length of 25 ms and a frame shift of 10 ms. Each feature vector was
concatenated with context data from the previous seven and subsequent seven
frames. Concatenated features are subsampled by a factor of ten. Consequently,
a $(23\times 15)$-dimensional input vector is fed into the neural network
every 100 ms.

As for the speaker embeddings, 512-dimensional ECAPA-TDNN xvectors with
different window sizes of 1 second, 2 and 3 seconds, and a window hop of 100
ms are used. Experiments with concatenated features have a dimension of 857
($23\times 15+512$), while in the others experiments, the input dimension is
the corresponding feature dimensionality. The time resolution remains constant
across all the input sequences.

We train our models using sim2spk dataset for 100 epochs. We employed the Adam
optimizer with the learning rate schedule and 200,000 warm-up steps as
recommended in [15]. Then we finetune these models to the 2 speakers “Adapt”
subset of CallHome for 50 epoch to adapt the models trained with simulated
data to real recordings.

###  3.3 System configurations

Table 2 shows the number of parameters for the models analyzed. For each
model, we use both the 3-encoder block-based model and the original 4-encoder
block, with the idea that integrating the more refined speaker embeddings with
respect to the acoustic features could lead to a simpler task for the encoder.
The extra parameters of the method proposed in section 2.3.1 comes from the
Emb Encoder transformer block that refines the speaker embedding sequence and
reduces its dimensionality before the EDA module.

Method | #Enc | #Parameters
---|---|---
| EEND-EDA
---
(baseline)
4 | 6.4M
Spk. Emb. into EDA | 3 | 6.5M
4 | 7.8M
Spk. Emb. into SA-EEND | 3 | 5.1M
4 | 6.4M
Concat. into SA-EEND | 3 | 5.2M
4 | 6.5M
Table 2: Number of parameters of the baseline and the proposed methods. #Enc
refers to the number of transformer encoder blocks in the SA-EEND encoder.

###  3.4 Metrics

To assess the effectiveness of the diarization process, we employed the
Diarization Error Rate (DER) as metric [46]. Consistent with established
methodologies, a collar of 0.25 seconds was permitted at the boundary of each
speech segment to account for potential time misalignments. DER, which
combines False Alarm Rate (FA), Missed Detection Rate (Miss) and Speaker Error
Rate (SE) , was calculated as the sum of these components divided by the total
speech duration in the reference annotation:

$DER=\frac{FA+Miss+SE}{Total\ Speech\ Duration}$

This metric provides a robust measure of system performance while considering
temporal variations around speech boundaries.

##  4 Results

Our first proposed method, described in section 2.3.1 and referred to as
“Speaker Embeddings into EDA Module,” investigates how speaker embeddings
influence the diarization performance of the EEND-EDA model when used as input
to the EDA module. Results from our experiments (see table 3) indicate that,
despite an increase in model parameters, incorporating speaker embeddings into
the EDA module does not enhance performance compared to the baseline EEND-EDA
model. This suggests that the speaker embeddings may not provide more
discriminative information when used in this manner within the diarization
task, than the one learned by the original EEND-EDA model.

|  | CH2-2spk
---|---|---
| EEND-EDA | 8.07 | - | -
#Enc | w/oracle | NoVAD | OraVAD | KaldiVAD
3 | ✗ | 20.11 | 17.96 | 19.31
✓ | 19.64 | 13.83 | 15.2
4 | ✗ | 18.44 | 16.75 | 17.29
✓ | 18.32 | 11.2 | 14.42
Table 3: Results in DER (%) on CH2-2spk of the Speaker embeddings into EDA module method, described in 2.3.1 with different system configurations, varying the number of encoder blocks, and whether the oracle VAD was used during training. The window size for extracting the embeddings is 1s. |  |  | CH2-2spk
---|---|---|---
| EEND-EDA |  | 8.07 | - | -
#Enc | Size | w/oracle | NoVAD | OraVAD | KaldiVAD
3 | 1s | ✗ | 26.77 | 23.21 | 24.96
✓ | 23.11 | 14.32 | 16.79
2s | ✗ | 26.2 | 19.43 | 21.68
✓ | 26.74 | 18.98 | 20.11
3s | ✗ | 31.18 | 21.72 | 26.98
✓ | 23.56 | 21.15 | 22.43
4 | 1s | ✗ | 32.3 | 24.28 | 28.05
✓ | 22.91 | 17.66 | 18.24
2s | ✗ | 28.1 | 20.82 | 21.35
✓ | 25.27 | 19.32 | 21.59
3s | ✗ | 30.2 | 21.39 | 27.4
✓ | 23.7 | 21.2 | 22.6
Table 4: Results in DER (%) on CH2-2spk for Speaker embeddings into SA-EEND encoder method, described in 2.3.2 with different system configurations, varying the number of the transformer encoder blocks, the window size for extracting the embeddings, and whether the oracle VAD was used during training. |  |  | CH2-2spk
---|---|---|---
|  |  | w/o adapt | w/adapt
| EEND-EDA (Baseline) |  | 10.09 | - | 8.07 | - | -
#Enc | Size | w/oracle | NoVAD | OracleVAD | NoVAD | OracleVAD | kaldiVAD
3 | 1s | ✗ | 19.23 | 14.42 | 12.87 | 8.4 | 8.79
✓ | 16.99 | 10.39 | 12.13 | 7.43 | 8.07
2s | ✗ | 17.33 | 13.81 | 12.32 | 8.64 | 8.8
✓ | 15.34 | 9.95 | 13.65 | 7.66 | 8.03
3s | ✗ | 17.16 | 16.87 | 9.68 | 9.01 | 9.07
✓ | 19.41 | 9.01 | 13.71 | 7.62 | 8.12
4 | 1s | ✗ | 21.7 | 13.43 | 12.55 | 8.34 | 8.53
✓ | 15.74 | 9.86 | 13.05 | 7.2 | 7.6
2s | ✗ | 16.4 | 13.93 | 11.74 | 8.45 | 8.49
✓ | 18.98 | 10.67 | 12.31 | 7.4 | 7.84
3s | ✗ | 15.7 | 14.9 | 9.12 | 8.8 | 9.09
✓ | 20.44 | 11.95 | 13.76 | 7.54 | 8.56
Table 5: Results in DER (%) on CH2-2spk for different system configurations of
the Concatenation of speaker embeddings and MFbank into SA-EEND encoder
described in 2.3.3, varying the number of encoder blocks, the window size used
to extract the embeddings, and whether the oracle VAD was used during the
model training.

Our second proposed method, explained in section 2.3.2 and referred to as
“Speaker Embeddings into SA-EEND Encoder,” evaluates the impact of utilizing
only these speaker embeddings as input to the SA-EEND encoder, and therefore
as the only information provided to the whole EEND-EDA model. Table 4 displays
our findings, revealing that a model configuration with 3 encoders and a
1-second window size for speaker embedding extraction yields the best results.
Employing this setup during EEND-EDA training, we achieve a DER of 14.32% when
using during evaluation the oracle VAD segmentation and 16.79% when applying
an external VAD. While these results do not surpass the baseline, this
approach demonstrates the significant potential of speaker embeddings as the
sole input for end-to-end diarization.

Figure 2: TSNE Visualization of embeddings for conventional SA-EEND-EDA and our best approach. a) Audio from simulated data, b) iafi recording, c) ialq recording. adapt | Method | VAD | DER
---|---|---|---
✗ | Baseline EEND-EDA | - | 10.09
Concat EEND-EDA (3 enc, 3 win len) | Oracle | 9.01
Concat EEND-EDA (4 enc, 1 win len) | Oracle | 9.86
✓ | Baseline EEND-EDA | - | 8.07
Concat EEND-EDA (4 enc, 1 win len) | Oracle | 7.2
Concat EEND-EDA (4 enc, 1 win len) | Kaldi | 7.6
Table 6: Results in DER (%) of the performing proposed systems that correspond
to the Concatenation of speaker embeddings and MFbank into SA-EEND encoder
method using different VAD strategy. The results in CallHome “test” set are
shown before and after finetuning to the CallHome “Adapt” subset.

###  4.1 Results of the concatenation of speaker embeddings with acoustic
features

In Table 5, the results obtained by the models using the proposed method
described in section 2.3.3 and so-called “Concatenation of speaker embeddings
and MFbank into SA-EEND encoder,” are presented.

The performance without finetuning the models to the CallHome “Adapt” subset
is very similar to the baseline, although experiments trained and evaluated
with oracle VAD segmentation, using 3 encoder blocks and a 3-second extraction
window, as well as with 4 encoder blocks and a 1-second extraction window,
achieve a DER of 9.01% and 9.86%. This represents a relative improvement over
the baseline (10.09% DER in this condition) of 10.7% and 2.27%, respectively.

When finetuning the models to real data using the CallHome “Adapt” subset, and
evaluating using oracle VAD segmentation, their performance is significantly
improved, surpassing the baseline. Specifically, the best-performing model
uses 4 encoder transformer blocks and extracts embeddings with a 1-second
sized window. It achieves a DER of 7.2%, while the baseline EEND-EDA has a
performance of 8.07% after adaptation to real conversations. This represents a
relative improvement of 10.78%. For a more realistic perspective, without
oracle VAD segmentation during inference and using instead the external
energy-based VAD, this model achieves a DER of 7.6%. The relative improvement
in this case reduces to 5.82%, highlighting the importance of a reliable VAD
for leveraging proper information from these embeddings in the diarization
task.

Experiments where oracle VAD segmentation was not used during training, were
not as satisfactory due to the silence variability leaking into the speaker
embeddings. In this case, the experiment yielding the best DER uses 4 encoder
layers and a 1-second window (the same configuration of the best model when
trained with oracle VAD segmentation). The results obtained evaluating the
model without oracle VAD is 12.55%, whereas with oracle VAD segmentation, it
is 8.34%. It should be noted that in all experiments, whether trained with
oracle VAD segmentation or not, an improvement is observed when using oracle
VAD or an external VAD at evaluation time. This indicates that the EEND-EDA
model’s ability to discern between speech and non-speech segments is quite
limited and that using external models to assist in this task improves the
overall performance. Additionally, our experiments confirm the hypothesis that
using smaller window sizes for speaker embedding extraction, while potentially
reducing the quality of individual speaker representations, it positively
improves the overall speaker diarization performance.

Finally, the comparison between the two best-performing systems, both adapted
and non-adapted, with their respective baselines is shown in table 6.

###  4.2 Visual Analysis

In Figure 2, the output vectors $e$ from the last SA-EEND transformer encoder
block of the baseline are shown on the left, and those from the best
performing of the proposed methods are shown on the right. The first row
displays a randomly chosen audio file from the simulated data development set.
The second row shows the iafi audio file of the CallHome dataset, which is the
most difficult audio to diarize for our model with a DER of 31.91%, having the
baseline a 48.52% DER for this audio. The last row shows the ialq audio file
with the best DER result using our model (0%), while the baseline provides a
1.64% DER. In the last two cases, it can be observed that, in the proposed
model, thanks to the use of an oracle VAD during training, the embeddings
corresponding to silence are grouped into a more distinct cluster than in the
baseline. Moreover, the distance between clusters appears to be greater, and
the intra-cluster distance, for the overlap class, seems to be smaller.

##  5 Conclusions

In this study, we explore the integration of speaker embeddings with acoustic
features for the diarization of two speakers in telephonic data within end-to-
end neural diarization approaches. Our findings indicate that incorporating
speaker information embeddings into the well-known EEND-EDA model leads to
substantial improvements in diarization performance, in particular when using
as input a concatenation of speaker embeddings with acoustic feature. Our
results also showcase the importance of correctly handling silence segments
and optimizing window sizes for extracting speaker information for the
diarization task. The use of speaker information enhances the model’s ability
to discriminate between speakers, while keeping the overlap handling
capabilities of end-to-end neural systems, leading to further improvements in
speaker diarization performance with respect to the EEND-EDA baseline.

## Acknowledgments

This work was supported by FPI RTI2018-098091-B-I00,
MCIU/AEI/10.13039/501100011033/FEDER, UE and PID2021-125943OB-I00,
MCIN/AEI/10.13039/501100011033/FEDER, UE from the Spanish Ministerio de
Ciencia e Innovacion, Agencia y del Fondo Europeo de Desarrollo Regional.

## References

  * [1] Tae Jin Park, Naoyuki Kanda, Dimitrios Dimitriadis, Kyu J. Han, Shinji Watanabe, and Shrikanth Narayanan,  “A review of speaker diarization: Recent advances with deep learning,”  Computer Speech and Language, vol. 72, 3 2022.
  * [2] Shuo-Yiin Chang, Bo Li, Gabor Simko, Tara N. Sainath, Anshuman Tripathi, Aäron van den Oord, and Oriol Vinyals,  “Temporal modeling using dilated convolution and gating for voice-activity-detection,”  in 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2018, pp. 5549–5553.
  * [3] Noor Salwani Ibrahim and Dzati Athiar Ramli,  “I-vector extraction for speaker recognition based on dimensionality reduction,”  Procedia Computer Science, vol. 126, pp. 1534–1540, 2018.
  * [4] Gregory Sell and Daniel Garcia-Romero,  “Speaker diarization with plda i-vector scoring and unsupervised calibration,”  in 2014 IEEE Spoken Language Technology Workshop (SLT), 2014, pp. 413–417.
  * [5] Li Wan, Quan Wang, Alan Papir, and Ignacio Lopez Moreno,  “Generalized end-to-end loss for speaker verification,”  in 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). 2018, p. 4879–4883, IEEE Press.
  * [6] Daniel Garcia-Romero, David Snyder, Gregory Sell, Daniel Povey, and Alan McCree,  “Speaker diarization using deep neural network embeddings,”  in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2017, pp. 4930–4934.
  * [7] Nauman Dawalatabad, Mirco Ravanelli, Franccois Grondin, Jenthe Thienpondt, Brecht Desplanques, and Hwidong Na,  “Ecapa-tdnn embeddings for speaker diarization,”  in Interspeech, 2021.
  * [8] Desh Raj, David Snyder, Daniel Povey, and Sanjeev Khudanpur,  “Probing the information encoded in x-vectors,”  2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pp. 726–733, 2019.
  * [9] Dimitrios Dimitriadis and Petr Fousek,  “Developing On-Line Speaker Diarization System,”  in Proc. Interspeech 2017, 2017, pp. 2739–2743.
  * [10] Quan Wang, Carlton Downey, Li Wan, Philip Andrew Mansfield, and Ignacio Lopz Moreno,  “Speaker diarization with lstm,”  in 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2018, pp. 5239–5243.
  * [11] Federico Landini, Ján Profant, Mireia Diez, and Lukáš Burget,  “Bayesian hmm clustering of x-vector sequences (vbx) in speaker diarization: theory, implementation and analysis on standard tasks,”  Computer Speech & Language, vol. 71, pp. 101254, 2022.
  * [12] Youngki Kwon, Hee-Soo Heo, Jee-Weon Jung, You Jin Kim, Bong-Jin Lee, and Joon Son Chung,  “Multi-scale speaker embedding-based graph attention networks for speaker diarisation,”  in ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2022, pp. 8367–8371.
  * [13] Yusuke Fujita, Shinji Watanabe, Shota Horiguchi, Yawen Xue, and Kenji Nagamatsu,  “End-to-end neural diarization: Reformulating speaker diarization as simple multi-label classification,”  ArXiv, vol. abs/2003.02966, 2020.
  * [14] Luca Serafini, Samuele Cornell, Giovanni Morrone, Enrico Zovato, Alessio Brutti, and Stefano Squartini,  “An experimental review of speaker diarization methods with application to two-speaker conversational telephone speech recordings,”  Computer Speech & Language, vol. 82, pp. 101534, 2023.
  * [15] Yusuke Fujita, Naoyuki Kanda, Shota Horiguchi, Kenji Nagamatsu, and Shinji Watanabe,  “End-to-end neural speaker diarization with permutation-free objectives,”  in Interspeech, 2019.
  * [16] Yusuke Fujita, Naoyuki Kanda, Shota Horiguchi, Yawen Xue, Kenji Nagamatsu, and Shinji Watanabe,  “End-to-end neural speaker diarization with self-attention,”  in 2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), 2019, pp. 296–303.
  * [17] Shota Horiguchi, Yusuke Fujita, Shinji Watanabe, Yawen Xue, and Kenji Nagamatsu,  “End-to-end speaker diarization for an unknown number of speakers with encoder-decoder based attractors,”  in Interspeech, 2020.
  * [18] Shota Horiguchi, Yusuke Fujita, Shinji Watanabe, Yawen Xue, and Paola Garcia,  “Encoder-decoder based attractors for end-to-end neural diarization,”  IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 30, pp. 1–1, 01 2022.
  * [19] Keisuke Kinoshita, Marc Delcroix, and Naohiro Tawara,  “Integrating end-to-end neural and clustering-based diarization: Getting the best of both worlds,”  in ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2021, pp. 7198–7202.
  * [20] Ye-Rin Jeoung, Joon-Young Yang, Jeong-Hwan Choi, and Joon-Hyuk Chang,  “Improving transformer-based end-to-end speaker diarization by assigning auxiliary losses to attention heads,”  in ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2023, pp. 1–5.
  * [21] Zhengyang Chen, Bing Han, Shuai Wang, and Yanmin Qian,  “Attention-based encoder-decoder network for end-to-end neural speaker diarization with target speaker attractor,”  in Proc. Interspeech 2023, 08 2023, pp. 3552–3556.
  * [22] Soumi Maiti, Yushi Ueda, Shinji Watanabe, Chunlei Zhang, Meng Yu, Shi-Xiong Zhang, and Yong Xu,  “Eend-ss: Joint end-to-end neural speaker diarization and speech separation for flexible number of speakers,”  in 2022 IEEE Spoken Language Technology Workshop (SLT), 2023, pp. 480–487.
  * [23] Tobias Cord-Landwehr, Christoph Boeddeker, Cătălin Zorilă, Rama Doddipatla, and Reinhold Haeb-Umbach,  “Frame-wise and overlap-robust speaker embeddings for meeting diarization,”  in ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2023, pp. 1–5.
  * [24] Wei Xia, Han Lu, Quan Wang, Anshuman Tripathi, Yiling Huang, Ignacio Lopez Moreno, and Hasim Sak,  “Turn-to-diarize: Online speaker diarization constrained by transformer transducer speaker turn detection,”  in Proc. ICASSP, 2022, pp. 8077–8081.
  * [25] Guanlong Zhao, Quan Wang, Han Lu, Yiling Huang, and Ignacio Lopez Moreno,  “Augmenting transformer-transducer based speaker change detection with token-level training loss,”  in ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2023, pp. 1–5.
  * [26] Ivan Medennikov, Maxim Korenevsky, Tatiana Prisyach, Yuri Khokhlov, Maria Korenevskaya, Ivan Sorokin, Tatiana Timofeeva, Anton Mitrofanov, Andrei Andrusenko, Ivan Podluzhny, Aleksandr Laptev, and Aleksei Romanenko,  “Target-speaker voice activity detection: A novel approach for multi-speaker diarization in a dinner party scenario,”  in Interspeech, 10 2020, pp. 274–278.
  * [27] Weiqing Wang and Ming Li,  “Incorporating end-to-end framework into target-speaker voice activity detection,”  in ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2022, pp. 8362–8366.
  * [28] Dongmei Wang, Xiong Xiao, Naoyuki Kanda, Takuya Yoshioka, and Jian Wu,  “Target speaker voice activity detection with transformers and its integration with end-to-end neural diarization,”  in ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2023, pp. 1–5.
  * [29] Yi Liu, Eunjung Han, Chul Lee, and Andreas Stolcke,  “End-to-end neural diarization: From transformer to conformer,”  in Interspeech, 08 2021, pp. 3081–3085.
  * [30] Dong Yu, Morten Kolbæk, Zheng-Hua Tan, and Jesper Jensen,  “Permutation invariant training of deep models for speaker-independent multi-talker speech separation,”  in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2017, pp. 241–245.
  * [31] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou,  “Arcface: Additive angular margin loss for deep face recognition,”  in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 4685–4694.
  * [32] Sergey Ioffe,  “Probabilistic linear discriminant analysis,”  in Computer Vision–ECCV 2006: 9th European Conference on Computer Vision, Graz, Austria, May 7-13, 2006, Proceedings, Part IV 9. Springer, 2006, pp. 531–542.
  * [33] Arsha Nagrani, Joon Son Chung, and Andrew Zisserman,  “Voxceleb: a large-scale speaker identification dataset,”  arXiv preprint arXiv:1706.08612, 2017.
  * [34] Joon Son Chung, Arsha Nagrani, and Andrew Zisserman,  “Voxceleb2: Deep speaker recognition,”  arXiv preprint arXiv:1806.05622, 2018.
  * [35] David Snyder, Guoguo Chen, and Daniel Povey,  “Musan: A music, speech, and noise corpus,”  10 2015.
  * [36] Desh Raj, David Snyder, Daniel Povey, and Sanjeev Khudanpur,  “Probing the information encoded in x-vectors,”  in 2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU). IEEE, 2019, pp. 726–733.
  * [37] Daniel Povey, Arnab Ghoshal, Gilles Boulianne, Lukas Burget, Ondrej Glembek, Nagendra Goel, Mirko Hannemann, Petr Motlicek, Yanmin Qian, Petr Schwarz, et al.,  “The kaldi speech recognition toolkit,”  in IEEE 2011 workshop on automatic speech recognition and understanding. IEEE Signal Processing Society, 2011, number CONF.
  * [38] Federico Landini, Alicia Lozano-Diez, Mireia Diez, and Lukáš Burget,  “From Simulated Mixtures to Simulated Conversations as Training Data for End-to-End Neural Diarization,”  in Proc. Interspeech 2022, 2022, pp. 5095–5099.
  * [39] J.J. Godfrey, E.C. Holliman, and J. McDaniel,  “Switchboard: telephone speech corpus for research and development,”  in [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, 1992, vol. 1, pp. 517–520 vol.1.
  * [40] Mark Przybocki and Alvin F. Martin,  “NIST speaker recognition evaluation chronicles,”  in Proc. The Speaker and Language Recognition Workshop (Odyssey 2004), 2004, pp. 15–22.
  * [41] Omid Sadjadi, Craig Greenberg, Elliot Singer, Lisa Mason, and Douglas Reynolds,  “Nist 2021 speaker recognition evaluation plan,” 2021-07-12 04:07:00 2021.
  * [42] Mark A. Przybocki, Alvin F. Martin, and Audrey N. Le,  “Nist speaker recognition evaluation chronicles - part 2,”  in 2006 IEEE Odyssey - The Speaker and Language Recognition Workshop, 2006, pp. 1–6.
  * [43] Alvin F Martin and Craig S Greenberg,  “Nist 2008 speaker recognition evaluation: Performance across telephone and room microphone channels,”  in Tenth Annual Conference of the International Speech Communication Association, 2009.
  * [44] Tom Ko, Vijayaditya Peddinti, Daniel Povey, Michael L. Seltzer, and Sanjeev Khudanpur,  “A study on data augmentation of reverberant speech for robust speech recognition,”  in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2017, pp. 5220–5224.
  * [45] Alvin Martin and Mark Przybocki,  “The nist 1999 speaker recognition evaluation—an overview,”  Digital signal processing, vol. 10, no. 1-3, pp. 1–18, 2000.
  * [46] Jonathan G Fiscus, Jerome Ajot, and John S Garofolo,  “The rich transcription 2007 meeting recognition evaluation,”  in International Evaluation Workshop on Rich Transcription. Springer, 2007, pp. 373–389.


# Unique Differences in Symmetric Subsets of $\mathbb{F}_{p}$

Tai Do Duc
Division of Mathematical Sciences
School of Physical & Mathematical Sciences
Nanyang Technological University
Singapore 637371
Republic of Singapore
Bernhard Schmidt
Division of Mathematical Sciences
School of Physical & Mathematical Sciences
Nanyang Technological University
Singapore 637371
Republic of Singapore

###### Abstract

Let $p$ be a prime and let $A$ be a subset of $\mathbb{F}_{p}$ with $A=-A$ and
$|A\setminus\\{0\\}|\leq 2\log_{3}(p)$. Then there is an element of
$\mathbb{F}_{p}$ which has a unique representation as a difference of two
elements of $A$.

##  1 Introduction

Let $p$ be a prime and let $\mathbb{F}_{p}$ denote the field with $p$
elements. Let $A$ be a nonempty subset of $\mathbb{F}_{p}$. We say that $A$
has a unique difference if there is $x\in\mathbb{F}_{p}$ such that there is
exactly one ordered pair $(a,b)$, $a,b\in A$, with $x=a-b$. Unique sums are
defined similarly. A subset $B$ of $\mathbb{F}_{p}$ is called symmetric if
$B=-B$.

According to [13], the following problem was first proposed by W. Feit.

######  Problem 1.1.

Given a prime $p$, what is the largest number $f(p)$ such that every subset of
$\mathbb{F}_{p}$ with at most $f(p)$ elements has a unique difference?

Let $\log_{n}(\cdot)$ denote the logarithm with base $n$. Straus [13] showed
$f(p)\geq 1+\log_{4}(p-1)$. This result was improved by Browkin, Divis, and
Schinzel [1] who obtained

$f(p)\geq\log_{2}p,$ (1)

which is the best known lower bound for $f(p)$. It is not known whether this
bound is asymptotically sharp.

In [13, Thm. 2], subsets of $\mathbb{F}_{p}$ were constructed which do not
have unique differences and are of cardinality $(2+o(1))\log_{3}(p)$. These
sets are symmetric. Thus [13, Thm. 2] implies

$g(p)\leq(2+o(1))\log_{3}(p)$ (2)

where $g(p)$ denotes the largest number such that every symmetric subset of
$\mathbb{F}_{p}$ with at most $g(p)$ elements has a unique difference.

Note that a symmetric subset of $\mathbb{F}_{p}$ has a unique difference if
and only if it has a unique sum. Thus we could as well formulate the results
of this paper in terms of unique sums.

The above-mentioned result of Browkin, Divis, and Schinzel implies

$g(p)\geq\log_{2}p.$ (3)

The following theorem is the main result of this paper. It implies $g(p)\geq
2\log_{3}(p)$, which is a substantial improvement upon (3).

######  Theorem 1.2.

Let $p\geq 5$ be a prime and let $A$ be a symmetric subset of $\mathbb{F}_{p}$
with $|A\setminus\\{0\\}|\leq 2\log_{3}(p)$. Then $A$ has a unique difference.

In view of Straus’ result (2), Theorem 1.2 is sharp in the following sense.

######  Corollary 1.3.

We have $g(p)\geq 2\log_{3}(p)$ for every prime $p$. Moreover, for every
$\varepsilon>0$, there exists a constant $C(\varepsilon)$ such that

$g(p)\leq(2+\varepsilon)\log_{3}(p)$ (4)

for every prime $p>C(\varepsilon)$.

Results like (1) and Theorem 1.2 have applications in various areas, see [1,
5, 6, 7, 8, 9, 10], for instance. In the last section, we present a new
application to cyclotomic integers $X$ for which $|X|^{2}$ is an integer.

##  2 Preliminaries

In this section, we state some well known results which will be needed later.
We include proofs for the convenience of the reader. For a ring $R$, let
$M_{m,n}(R)$ denote the set of $m\times n$ matrices with entries from $R$. The
Euclidean norm of $x\in\mathbb{R}^{n}$ is denoted by $||x||$.

######  Result 2.1.

Let $m\leq n$ and let $A\in M_{m,n}(\mathbb{R})$ with rows
$r_{1},\ldots,r_{m}$. Set $d(1)=||r_{1}||$. For $2\leq j\leq m$, let $d(j)$ be
the distance of $r_{j}$ from the subspace of $\mathbb{R}^{n}$ spanned by
$r_{1},\ldots,r_{j-1}$. We have

$\det(AA^{T})=\prod_{j=1}^{m}d(j)^{2}.$ (5)

###### Proof.

If ${\rm rank}_{\mathbb{R}}(A)<m$, then left hand and right side of (5) are
both zero. Hence we may assume ${\rm rank}_{\mathbb{R}}(A)=m$. By Gram-Schmidt
orthogonalization, there is a nonsingular lower triangular matrix $L\in
M_{m\times m}$ with diagonal entries $d(j)^{-1}$, $j=1,\ldots,m$, such that
the rows of $Q=LA$ are an orthonormal basis of the subspace of
$\mathbb{R}^{n}$ spanned by $r_{1},\ldots,r_{m}$. Thus $QQ^{T}=I_{m}$ where
$I_{m}$ denotes the $m\times m$ identity matrix. We conclude
$AA^{T}=L^{-1}QQ^{T}(L^{-1})^{T}=L^{-1}(L^{-1})^{T}$. Hence

$\det(AA^{T})=(\det(L^{-1}))^{2}=\prod_{j=1}^{m}d(j)^{2}.$

∎

######  Result 2.2.

Let $A\in M_{u,n}(\mathbb{R})$, $B\in M_{w,n}(\mathbb{R})$, and $C={A\choose
B}$. Then

$\det(CC^{T})\leq\det(AA^{T})\det(BB^{T}).$

###### Proof.

Let $r_{1},\ldots,r_{u}$ be the rows of $A$ and let $r_{u+1},\ldots,r_{u+w}$
be the rows of $B$. For $1\leq i<j\leq u+w$, let $d(i,j)$ denote the distance
of $r_{j}$ from the subspace of $\mathbb{R}^{n}$ generated by
$r_{i},\ldots,r_{j-1}$. Furthermore, set $d(i,i)=||r_{i}||$ for all $i$. By
Result 2.1, we have

$\displaystyle\det(AA^{T})$ $\displaystyle=\prod_{j=1}^{u}d(1,j)^{2},$
$\displaystyle\det(BB^{T})$ $\displaystyle=\prod_{j=u+1}^{u+w}d(u+1,j)^{2},$
$\displaystyle\det(CC^{T})$ $\displaystyle=\prod_{j=1}^{u+w}d(1,j)^{2}.$

Moreover, $d(1,j)\leq d(u+1,j)$ for $j\geq u+1$ by the definition of the
$d(i,j)$’s. Hence

$\displaystyle\det(CC^{T})$ $\displaystyle=\prod_{j=1}^{u+w}d(1,j)^{2}$
$\displaystyle\leq\prod_{j=1}^{u}d(1,j)^{2}\prod_{j=u+1}^{u+w}d(u+1,j)^{2}$
$\displaystyle=\det(AA^{T})\det(BB^{T}).$

∎

A repeated application of Result 2.2 gives the following.

######  Corollary 2.3.

Let $A_{i}\in M_{u_{i},n}(\mathbb{R})$, $i=1,\ldots,k$, and

$C=\begin{pmatrix}A_{1}\\\ A_{2}\\\ \vdots\\\ A_{k}\end{pmatrix}.$

Then

$\det(CC^{T})\leq\prod_{i=1}^{k}\det(A_{i}A_{i}^{T}).$

##  3 Set-up

In this section, we introduce some notation and assumptions which will be
implicitly assumed in the rest of the paper. Let $p$ be an odd prime and
suppose that $A$ is a symmetric subset of $\mathbb{F}_{p}$ which has no unique
difference.

Recall that we assume $p\geq 5$. Write $A=\left\\{\pm a_{1},...,\pm
a_{n}\right\\}$ such that $a_{j}\neq\pm a_{i}$ for $i\neq j$. We set $a_{n}=0$
if $0\in A$. Let $m=\lfloor|A|/2\rfloor$. Note that $m=n$ if $0\not\in A$ and
$m=n-1$ if $0\in A$.

Next, we show that we may assume $|A|\geq 4$. If $|A|=1$, then $A$ certainly
has a unique difference. If $2\leq|A|\leq 3$, then $A=\\{\pm a_{1}\\}$ or
$A=\\{0,\pm a_{1}\\}$ for some $a_{1}\neq 0$. Then $a_{1}-(-a_{1})=2a_{1}$ is
a unique difference in $A$. Hence we can indeed assume $|A|\geq 4$.

We now set up a linear system arising from $A$ and derive some useful
properties of its coefficient matrix. Let $i$ be arbitrary with $1\leq i\leq
m$. As $2a_{i}=a_{i}-(-a_{i})$ is not a unique difference in $A$ and
$2a_{i}\neq\pm a_{i}-a_{i}$, there exists an ordered pair
$(\sigma(i),\tau(i))\neq(i,i)$ with

$2a_{i}\pm a_{\sigma(i)}\pm a_{\tau(i)}=0.$ (6)

Here “$\pm a_{\sigma(i)}\pm a_{\tau(i)}$” means that any combination of signs
is possible including $a_{\sigma(i)}-a_{\tau(i)}$ and
$-a_{\sigma(i)}+a_{\tau(i)}$. We use this convention throughout the rest of
the paper.

We consider the homogeneous linear system corresponding to these equations:

$2x_{i}\pm x_{\sigma(i)}\pm x_{\tau(i)}=0,\ i=1,\ldots,m.$ (7)

Here we use the convention $x_{n}=0$ if $0\in A$ (and thus $a_{n}=0$).

Note that the coefficient vectors corresponding to the system (7) all have at
most $3$ nonzero entries. The nonzero coefficients, however, are not
necessarily $2,\pm 1$, since $i,\sigma(i),\tau(i)$ are not necessarily
distinct. We now determine exactly which coefficient vectors can occur.

Case 1 $\sigma(i)=i$ or $\tau(i)=i$. By symmetry, we can assume $\tau(i)=i$.
Thus $2a_{i}\pm a_{\sigma(i)}\pm a_{i}=0$. As $(\sigma(i),\tau(i))\neq(i,i)$,
we have $\sigma(i)\neq i$. If $2a_{i}\pm a_{\sigma(i)}-a_{i}=0$, then
$a_{i}\pm a_{\sigma(i)}=0$, contradicting the assumption $a_{i}\neq\pm a_{j}$
for $i\neq j$. Hence $2a_{i}\pm a_{\sigma(i)}+a_{i}=3a_{i}\pm
a_{\sigma(i)}=0$. Recall that we assume $p\geq 5$. If $a_{\sigma(i)}=0$, then
$3a_{i}=0$ and thus $a_{i}=0$, contradicting the assumptions. Thus
$a_{\sigma(i)}\neq 0$, i.e., $\sigma(i)\leq m$.

Hence (6) can be written as

$3x_{i}\pm x_{\sigma(i)}=0$ (8)

with $\sigma(i)\neq i$ and $\sigma(i)\leq m$. We call (8) an equation of type
1.

Case 2 $\sigma(i)\neq i$ and $\tau(i)\neq i$. If $\tau(i)=\sigma(i)$, then (6)
implies $a_{i}=0$ or $a_{i}=\pm a_{\sigma(i)}$, contradicting the assumptions.
Thus $\tau(i)\neq\sigma(i)$. Hence $i,\sigma(i),\tau(i)$ are pairwise
distinct.

Recall that $x_{n}=0$ if $0\in A$. Hence, if $0\in A$ and
$n\in\\{\sigma(i),\tau(i)\\}$, then one of the variables
$x_{\sigma(i)},x_{\tau(i)}$ occurs with coefficient zero in (6). By symmetry,
we can assume that $x_{\tau(i)}$ occurs with coefficient zero in this case.

Hence, in Case 2, we can write (6) as

$\begin{split}2x_{i}\pm x_{\sigma(i)}=0&\text{\ \ if $0\in A$ and
$\tau(i)=n$},\\\ 2x_{i}\pm x_{\sigma(i)}\pm x_{\tau(i)}=0&\text{\ \
otherwise,}\end{split}$ (9)

where $i,\sigma(i),\tau(i)$ are pairwise distinct. In both cases, we call (9)
an equation of type 2.

In summary, as Case 1 and Case 2 cover all possible cases, for every
$i\in\\{1,\ldots,m\\}$, one of the following equations is contained in the
linear system (7).

$\begin{split}3x_{i}\pm x_{\sigma(i)}=0&\text{\ \ (type 1)},\\\ 2x_{i}\pm
x_{\sigma(i)}=0&\text{\ \ (type 2)},\\\ 2x_{i}\pm x_{\sigma(i)}\pm
x_{\tau(i)}=0&\text{\ \ (type 2).}\end{split}$

Furthermore, the following hold.

  * •

$\sigma(i)\leq m$ and $\tau(i)\leq m$,

  * •

$i,\sigma(i),\tau(i)$ are pairwise distinct.

Of course, the statements involving $\tau(i)$ only apply if the equation
$2x_{i}\pm x_{\sigma(i)}\pm x_{\tau(i)}=0$ is contained (7).

We use a similar terminology for the coefficient vectors of the system (7): We
say a coefficient vector is of type 1 if it has exactly one entry $3$, exactly
one entry $\pm 1$, and all its remaining entries are zero. A coefficient
vector is of type 2 if it has exactly one entry $2$, at most two entries $\pm
1$, and all its remaining entries are zero.

##  4 A Congruence for Minors of the Coefficient Matrix

Let $M$ be the coefficient matrix of the linear system (7). Recall that all
entries of $M$ are from $\\{0,\pm 1,2,3\\}$. Note that $M$ can be considered
as a matrix with rational entries as well as a matrix with entries from
$\mathbb{F}_{p}$. In the following, we make use of both interpretations. Let
$r$ be the rank of the coefficient matrix $M$ over $\mathbb{Q}$. We now prove
that all $r\times r$-minors of $M$ are divisible by $p$. This result is
useful, as it can be combined with estimates for minors of $M$ which we get
from Result 2.1.

######  Theorem 4.1.

Suppose that $A=\\{\pm a_{1},...,\pm a_{n}\\}\subset\mathbb{F}_{p}$ does not
have a unique difference and let $M$ be the coefficient matrix of the
corresponding linear system $(\ref{system})$. Write $r={\rm
rank}_{\mathbb{Q}}(M)$. Every $r\times r$-minor of $M$ is divisible by $p$.

###### Proof.

Recall that $M$ is an $m\times m$-matrix and that $M$ has entries from
$\\{0,\pm 1,2,3\\}$ only. In this proof, we consider the entries of $M$ as
integers (not as elements of $\mathbb{F}_{p}$), and we will work with the
Smith Normal Form of $M$ over the integers.

Let $D=\text{diag}(d_{1},...,d_{r},0,...,0)$ be the Smith Normal Form of $M$.
Note that $d_{1},\ldots,d_{r}$ are integers such that $d_{i}$ divides
$d_{i+1}$ for $i=1,\ldots,r-1$. Furthermore, there are $B,C\in
M_{m,m}(\mathbb{Z})$ with $\det(B)=\pm 1$, $\det(C)=\pm 1$, and $M=BDC$. It is
important for this proof to note that $B$, $C$, $D$ are matrices with integer
entries and the equation $M=BDC$ holds over $\mathbb{Z}$ (not only over
$\mathbb{F}_{p}$). Furthermore, note that all entries of $C^{-1}$ are
integers, as $\det(C)=\pm 1$.

Write $x=(a_{1},\ldots,a_{m})^{T}$. Note $x\in\mathbb{F}_{p}^{m}$. We have
$x\neq 0$, as $|A|>1$. Recall that $Mx=0$ over $\mathbb{F}_{p}$ by (7).

Suppose that $p$ does not divide $d_{r}$. Then $p$ does not divide any of the
integers $d_{1},\ldots,d_{r}$. Write $Cx=(b_{1},\ldots,b_{m})^{T}$ where
$b_{1},\ldots,b_{m}\in\mathbb{F}_{p}$. Note that $DCx=0$ over $\mathbb{F}_{p}$
if and only if $b_{1}=\cdots=b_{r}=0$, since we assume that
$d_{1},\ldots,d_{r}$ are not divisible by $p$. Hence $DCx=0$ if and only if
$x=C^{-1}(0,\ldots,0,b_{r+1},\ldots,b_{m})^{T}$ for some
$b_{i}\in\mathbb{F}_{p}$. As $BDCx=Mx=0$ over $\mathbb{F}_{p}$, we indeed have
$DCx=0$ over $\mathbb{F}_{p}$. Thus

$x=C^{-1}b\text{ with
}b=(0,\ldots,0,b_{r+1},\ldots,b_{m})^{T}\in\mathbb{F}_{p}^{m}.$ (10)

For convenience, we set $b_{i}=0$ for $i=1,\ldots,r$.

We now switch from equations over $\mathbb{F}_{p}$ to equations over
$\mathbb{Z}$. We first define a vector $c$ with integer entries which
represent the residue classes $b_{1},\ldots,b_{m}$ mod $p$. Formally, let
$c_{1},\ldots,c_{m}$ be the unique integers with $0\leq c_{i}\leq p-1$ and

$b_{i}=c_{i}+p\mathbb{Z}/\mathbb{Z}$ (11)

(here we use the standard notation
$\mathbb{F}_{p}=\\{k+p\mathbb{Z}/\mathbb{Z}:k=0,\ldots,p-1\\}$). Note that
$c_{1}=\cdots=c_{r}=0$, as $b_{1}=\cdots=b_{r}=0$ (again, note that
$c_{1}=\cdots=c_{r}=0$ are equations over $\mathbb{Z}$, not only over
$\mathbb{F}_{p}$).

Define $y=(y_{1},\ldots,y_{m})\in\mathbb{Z}^{m}$ by

$y=C^{-1}(0,\ldots,0,c_{r+1},\ldots,c_{m})^{T}$ (12)

(recall that the $c_{i}$’s are considered as integers, not as elements of
$\mathbb{F}_{p}$). Then we have

$My=BDCy=B\,\text{diag}(d_{1},...,d_{r},0,...,0)(0,\ldots,0,c_{r+1},\ldots,c_{m})^{T}=0$
(13)

over $\mathbb{Z}$ (note that all entries of matrices and vectors occurring in
(13) are considered as integers and to derive (13), we need the fact that the
equation $M=BDC$ holds over $\mathbb{Z}$, not only over $\mathbb{F}_{p}$).

Let $\Gamma_{j}$ be the $j$th row of $C^{-1}$, $j=1,\ldots,m$. By (10) and
(12), we have

$a_{j}=\Gamma_{j}b\text{ and }y_{j}=\Gamma_{j}c$ (14)

where $b=(0,\ldots,0,b_{r+1},\ldots,b_{m})^{T}$ and
$c=(0,\ldots,0,c_{r+1},\ldots,c_{m})^{T}$. Note that (11) and (14) imply

$a_{j}=y_{j}+p\mathbb{Z}/\mathbb{Z},\ i=1,\ldots,m.$ (15)

Recall that $M$ is the coefficient matrix of the linear system (7) and that
$My=0$ over $\mathbb{Z}$ by (13). This implies that $\\{\pm y_{1},\ldots,\pm
y_{m}\\}$ does not have a unique difference (where the differences are taken
in $\mathbb{Z}$). Recall that $a_{i}\neq\pm a_{j}$ for $i\neq j$ by
assumption. Thus (15) implies $y_{i}\neq\pm y_{j}$ for $i\neq j$. Hence there
is $k\in\\{1,\ldots,m\\}$ such that $|y_{k}|>|y_{i}|$ for all $i\neq k$. This
implies that $2y_{k}=y_{k}-(-y_{k})$ is a unique difference of $\\{\pm
y_{1},\ldots,\pm y_{m}\\}$, a contradiction. We conclude that $p$ divides
$d_{r}$.

From the theory of the Smith Normal Form (see [11, p. 41], for instance), it
is well known that $d_{r}$ divides the greatest common divisor of all $r\times
r$-minors of $M$. ∎

##  5 Equations of Type 1

Equations of type 1 play a critical role in the proof of Theorem 1.2, as the
Euclidean norm of their coefficient vectors is the largest among the equations
occurring in the linear system (7). In this section, we study the structure of
the set of equations of type 1 contained in (7). Recall that equations of type
1 have the form

$3x_{i}\pm x_{\sigma(i)}=0$

where $1\leq i\leq m$, $\sigma(i)\neq i$, and $\sigma(i)\leq m$.

######  Lemma 5.1.

Suppose $3^{\lfloor|A|/2\rfloor}\leq p$. Let $I$ be a subset of
$\\{1,\ldots,m\\}$ such that $3x_{i}\pm x_{\sigma(i)}=0$, $i\in I$, are
equations of type $1$ contained in $(\ref{system})$. Let $G$ be the directed
graph with vertex set $I\cup\\{\sigma(i):i\in I\\}$ and edge set
$E=\\{(i,\sigma(i)):i\in I\\}$. Then $E$ can be decomposed into directed paths
which are pairwise vertex disjoint.

###### Proof.

We show
(i) every vertex of $G$ has outdegree at most $1$,
(ii) every vertex of $G$ has indegree at most $1$,
(iii) $G$ does not contain a directed cycle.

Note that (i-iii) imply that $G$ indeed can be decomposed into directed paths
which are pairwise vertex disjoint.

First of all, there is at most one edge $(i,\sigma(i))$ for every vertex $i$.
This means that the outdegree of every vertex in $G$ is at most one.

Suppose that the indegree of a vertex $i$ is at least $2$. Then there are
distinct vertices $j,k$ with $\sigma(j)=\sigma(k)=i$. By definition, this
implies $3a_{j}\pm a_{i}=0$ and $3a_{k}\pm a_{i}=0$ and thus
$3(a_{j}-a_{k})=0$ or $3(a_{j}+a_{k})=0$. As $p>3$, we conclude $a_{j}=\pm
a_{k}$ which contradicts the assumption $a_{j}\neq\pm a_{k}$ for $j\neq k$.
This shows that all vertices of $G$ have indegree at most $1$.

Now suppose that $G$ contains a directed cycle. Then there are vertices
$v_{1},v_{2},\ldots,v_{k}$ with $v_{k}=v_{1}$ such that $(v_{i},v_{i+1})\in E$
for $i=1,\ldots,k-1$, i.e., $v_{i+1}=\sigma(v_{i})$ for $i=1,\ldots,k-1$. By
definition, this implies $3a_{v_{i}}=\pm a_{v_{i+1}}$ for $i=1,\ldots,k-1$.
Hence

$a_{v_{1}}=a_{v_{k}}=\pm 3a_{v_{k-1}}=\pm 9a_{v_{k-2}}=\cdots=\pm
3^{k-1}a_{v_{1}}.$

Thus $(\pm 3^{k-1}-1)a_{v_{1}}=0$. Note that $k\leq m+1$, as the cycle
contains every vertex $\neq v_{1}$ at most once. Moreover, $p\geq
3^{m}=3^{\lfloor|A|/2\rfloor}$ by assumption and thus $p\geq 3^{m}+2$, as
$p>3$ is a prime. We conclude $3^{k-1}+1\leq 3^{m}+1<p$. Hence $(\pm
3^{k-1}-1)a_{v_{1}}=0$ implies $a_{v_{1}}=0$, contradicting the assumption
$a_{i}\neq 0$ for $i=1,\ldots,m$. This shows that $G$ does not contain a
directed cycle.

In summary, we have shown that (i)-(iii) hold, and this completes the proof. ∎

######  Remark 5.2.

Note that Lemma 5.1 implies that $G$ does not contain any cycle, directed or
undirected.

Next, we compute determinants arising from matrices whose rows are coefficient
vectors of equations of type 1.

######  Lemma 5.3.

Let $G$ be the graph defined in Lemma 5.1. Suppose $J$ is a subset of
$\\{1,\ldots,m\\}$ such that $\\{(i,\sigma(i)):i\in J\\}$ is a directed path
in $G$ and let $B$ be the coefficient matrix of the corresponding equations
$3x_{i}\pm x_{\sigma(i)}=0$, $i\in J$. Then

$\det(BB^{T})=\frac{1}{8}\left(-1+9^{|J|+1}\right).$ (16)

###### Proof.

Write $v=|J|$. By relabeling vertices of $G$, if necessary, we may assume that
the directed path consists of the edges $(i,i+1)$, $i=1,\ldots,v$. The
corresponding equations are $3x_{i}\pm x_{i+1}=0$, $i=1,\ldots,v$. Hence $B$
is a $v\times m$-matrix of the form

$\begin{pmatrix}3&\pm 1&0&\cdots&0&0&0&\cdots&0\\\ 0&3&\pm
1&\cdots&0&0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&0&\cdots&0\\\ 0&0&0&\cdots&3&\pm
1&0&\cdots&0\\\ \end{pmatrix}.$

Thus

$BB^{T}=\begin{pmatrix}10&3\delta_{1}&0&\cdots&0&0&0\\\
3\delta_{1}&10&3\delta_{2}&\cdots&0&0&0\\\
0&3\delta_{2}&10&3\delta_{3}&\cdots&0&0\\\
\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\\
0&\cdots&0&3\delta_{v-3}&10&3\delta_{v-2}&0\\\
0&\cdots&0&0&3\delta_{v-2}&10&3\delta_{v-1}\\\
0&\cdots&0&0&0&3\delta_{v-1}&10\end{pmatrix}$

with $\delta_{i}=\pm 1$, $i=1,\ldots,v-1$. We now prove (16) by induction on
$v$. It is straightforward to check that (16) holds for $v=1$. Suppose $v\geq
2$. Using Laplace expansion with respect to the last column of $BB^{T}$ and
the inductive hypothesis, we find

$\det(BB^{T})=10\left(\frac{1}{8}\left(-1+9^{v}\right)\right)-(3\delta_{v-1})^{2}\left(\frac{1}{8}\left(-1+9^{v-1}\right)\right).$

Note $(3\delta_{v-1})^{2}=9$. Hence

$\det(BB^{T})=\frac{1}{8}\left(-10+9+(10-1)9^{v}\right)=\frac{1}{8}\left(-1+9^{v+1}\right).$

∎

######  Corollary 5.4.

Suppose $3^{\lfloor|A|/2\rfloor}\leq p$. Let $I$ be a subset of
$\\{1,\ldots,m\\}$ such that $3x_{i}\pm x_{\sigma(i)}=0$, $i\in I$, are
equations of type $1$. Let $G$ be the directed graph defined in Lemma 8 and
let $P_{1},\ldots,P_{t}$ be vertex disjoint directed paths in $G$ with
$E=\displaystyle\bigcup_{j=1}^{t}P_{j}$. Let $B$ be the matrix whose rows are
the coefficient vectors of the equations $3x_{i}\pm x_{\sigma(i)}=0$, $i\in
I$. Then

$\det(BB^{T})<11^{t}9^{|I|-t}.$

###### Proof.

For $j=1,\ldots,t$, let $B_{j}$ be the matrix whose rows are the coefficient
vectors of the equations $3x_{i}\pm x_{\sigma(i)}=0$, $(i,\sigma(i))\in
P_{j}$. By Lemma 5.3, we have

$\det(B_{j}B_{j}^{T})=\frac{1}{8}\left(-1+9^{|P_{j}|+1}\right).$

Note that

$\frac{1}{8}\left(-1+9^{x+1}\right)<11\cdot 9^{x-1}$

for all $x\geq 1$. Using Corollary 2.3 and $\sum_{j=1}^{t}|P_{j}|=|I|$, we
conclude

$\displaystyle\det(BB^{T})$
$\displaystyle\leq\prod_{j=1}^{t}\det(B_{j}B_{j}^{T})$
$\displaystyle=\prod_{j=1}^{t}\left(\frac{1}{8}\left(-1+9^{|P_{j}|+1}\right)\right)$
$\displaystyle<\prod_{i=1}^{t}\left(11\cdot 9^{|P_{j}|-1}\right)$
$\displaystyle=11^{t}9^{|I|-t}.$

∎

##  6 Proof of Theorem 1.2

Let $p$ be prime and let $A$ be a symmetric subset of $\mathbb{F}_{p}$ with

$|A\setminus\\{0\\}|\leq 2\log_{3}(p).$ (17)

We have to prove that $A$ has a unique difference. As shown in Section 3, we
may assume $p\geq 5$.

Suppose $A$ does not have a unique difference. Write $A=\\{\pm
a_{1},\ldots,\pm a_{n}\\}$ as in Section 3 where $a_{n}=0$ if $0\in A$. Recall
$m=\lfloor A/2\rfloor$. Note $|A\setminus\\{0\\}|=2\lfloor A/2\rfloor$. Hence
$3^{\lfloor A/2\rfloor}\leq p$ by (17) and thus

$3^{\lfloor A/2\rfloor}<p,$ (18)

as $p\neq 3$.

Let $M$ be the coefficient matrix of the linear system (7) and write $r={\rm
rank}_{\mathbb{Q}}(M)$. Note $r\leq m$. Let $N$ be a nonsingular $r\times r$
submatrix of $M$.

We claim that

$|\det(N)|\leq 3^{r}.$ (19)

We say that a row vector is of type 3 if it has exactly one entry $3$ and all
its remaining entries are zero. Note that $N$ may contain rows of type 3, as
some rows of $M$ of type 1 may turn into type 3 when columns of $M$ are
deleted. If $N$ contains a row of type 3, then, by Laplace expansion,
$|\det(N)|=3|\det(N_{1})|$ where $N_{1}$ is a $(r-1)\times(r-1)$ submatrix of
$M$. Repeating this process, if necessary, we either get $|\det(N)|=3^{r}$ or
obtain a $d\times d$ submatrix $N_{2}$ of $M$ with

$|\det(N)|=3^{r-d}|\det(N_{2})|$ (20)

such that $N_{2}$ does not contain any row of type 3. If $|\det(N)|=3^{r}$,
then (19) holds. Thus we may assume that (20) holds.

As $N_{2}$ is a submatrix of $M$, all rows of $N_{2}$ which are not of type 1
have at most three nonzero entries $\pm 1$, $2$, at most one of which is $2$.
Hence every row of $N_{2}$ which is not of type 1 has Euclidean norm at most
$\sqrt{6}$.

Swapping rows, if necessary, we can write

$N_{2}={B\choose C}$

where $B$ consists of the rows of $N_{2}$ of type $1$ and $C$ of the remaining
rows of $N_{2}$. Let $I$ be the subset of $\\{1,\ldots,m\\}$ such that
$3x_{i}\pm x_{\sigma(i)}=0$, $i\in I$, are the equations corresponding to the
rows of $B$. Note that $|I|$ is the number of rows of $B$ and $d-|I|$ is the
number of rows of $C$.

Let $G$ be the directed graph with vertex set $V=I\cup\\{\sigma(i):i\in I\\}$
and edge set $E=\\{(i,\sigma(i)):i\in I\\}$. Note that $|V|\leq d$, as every
element of $V$ is an index of a column of $N_{2}$.

By Lemma 5.1, there is a decomposition of $E$ into pairwise vertex disjoint
directed paths. Note that these directed paths are connected components of
$G$. Let $t$ be the number of paths in the decomposition. As $G$ has at most
$d$ vertices and contains no cycles, it has at most $|V|-|E|$ connected
components. Hence

$t\leq|V|-|E|\leq d-|I|.$ (21)

By Corollary 5.4, we have

$\det(BB^{T})<11^{t}9^{|I|-t}.$ (22)

From (21) and (22), we get

$\det(BB^{T})<11^{d-|I|}9^{|I|-(d-|I|)}=11^{d-|I|}9^{2|I|-d}.$ (23)

As the rows of $C$ all have Euclidean norm at most $\sqrt{6}$, we have

$\det(CC^{T})\leq 6^{d-|I|}$ (24)

by Result 2.1. From Result 2.2, we get

$\det(N_{2}N_{2}^{T})\leq\det(BB^{T})\det(CC^{T}).$ (25)

Putting (23-25) together, we find

$\displaystyle\det(N_{2}N_{2}^{T})$
$\displaystyle<11^{d-|I|}9^{2|I|-d}6^{d-|I|}$
$\displaystyle=66^{d-|I|}9^{2|I|-d}$ $\displaystyle\leq 81^{d-|I|}9^{2|I|-d}$
$\displaystyle=9^{d}.$

Hence $|\det(N_{2})|\leq 3^{d}$ and thus $|\det(N)|\leq 3^{r}$ by (20). This
proves (19).

Finally, recall that $\det(N)$ is a nonzero $r\times r$-minor of $M$. Hence
$|\det(N)|\geq p$ by Theorem 4.1. But $r\leq\lfloor|A|/2\rfloor$ and thus
$|\det(N)|\leq 3^{r}<p$ by (18), a contradiction. This completes the proof of
Theorem 1.2. $\Box$

##  7 Application to Weil Numbers

Let $m,n$ be positive integers. Write $\zeta_{m}=\exp(2\pi i/m)$. An $n$-Weil
number in $\mathbb{Z}[\zeta_{m}]$ is an element $Y$ of $\mathbb{Z}[\zeta_{m}]$
with $|Y|^{2}=n$. In this section, we show that Theorem 1.2 implies that under
certain conditions Weil numbers in $\mathbb{Z}[\zeta_{m}]$ necessarily are
contained in proper subfields of $\mathbb{Q}(\zeta_{m})$. This result is a
partial improvement of the “field descent” introduced in [12] and is relevant
for the study of difference sets and related objects. We will assume basic
algebraic number theory in this sections, as treated in [4], for instance.

For a finite group $G$ and a ring $R$, let $R[G]$ denote the group ring of $G$
over $R$. Every element $B$ of $R[G]$ can be written as $B=\sum_{g\in
G}r_{g}g$ with $r_{g}\in R$. The $r_{g}$’s are called the coefficients of $B$.
We write $B^{(-1)}=\sum_{g\in G}\overline{r_{g}}g^{-1}$ where
$\overline{r_{g}}$ is the complex conjugate of $r_{g}$.

For $Y\in\mathbb{Z}[\zeta_{m}]$, let

${\cal M}(Y)=\frac{1}{\varphi(m)}\sum_{\sigma\in{\rm
Gal}(\mathbb{Q}(\zeta_{m})/\mathbb{Q})}(Y\overline{Y})^{\sigma},$

where $\varphi$ denote the Euler totient function. Note

${\cal M}(Y)\geq 1$ (26)

for $Y\neq 0$ by the inequality of geometric and arithmetic means, since
$\prod{(Y\overline{Y})^{\sigma}}$ is the norm of an algebraic integer and thus
$\prod{(Y\overline{Y})^{\sigma}}\geq 1$. The following is due to Cassels [3].

######  Result 7.1.

Let $X\in\mathbb{Z}[\zeta_{m}]$ where $m=pm^{\prime}$ and $p$ is a prime with
$(p,m^{\prime})=1$. Write $X=\sum_{i=0}^{p-1}X_{i}\zeta_{p}^{i}$ with
$X_{i}\in\mathbb{Z}[\zeta_{m^{\prime}}]$. We have

$(p-1){\cal M}(X)=\sum_{i<j}^{p-1}{\cal M}(X_{i}-X_{j}).$

We denote the cyclic group of order $k$ by $C_{k}$.

######  Theorem 7.2.

Let $p$, $q$ be distinct primes and $r$ be a positive integer with
$\gcd(r,pq)=1$. Let $n=q^{b}$ where $b$ is a positive integer. Suppose that
$Y\bar{Y}=n$ for some $Y\in\mathbb{Z}[\zeta_{pr}]$. If ${\rm ord}_{p}(q)$ is
even and $p>\max\\{3^{n/2},n^{2}+n+1\\}$, then $Y\eta\in\mathbb{Z}[\zeta_{r}]$
for some root of unity $\eta$.

###### Proof.

Write ${\rm ord}_{p}(q)=2f$, and define $\sigma\in{\rm
Gal}(\mathbb{Q}(\zeta_{pr})/\mathbb{Q})$ by
$\zeta_{pr}^{\sigma}=\zeta_{pr}^{q^{f}}$. Note that $\sigma$ fixes all prime
ideals above $n$ in $\mathbb{Z}(\zeta_{pr})$ (see [12, Thm. 2.1], for
instance). Hence $Y^{\sigma}=Y\alpha$ for some unit $\alpha$. Note
$|\alpha|^{2}=Y^{\sigma}\overline{Y^{\sigma}}/(Y\overline{Y})=n^{\sigma}/n=1$.
Thus $\alpha$ is a root of unity, i.e., $\alpha=\pm\zeta_{p}^{c}\zeta_{r}^{d}$
for some integers $c,d$. Let $e$ be an integer with $2e\equiv c\pmod{p}$. Note
that $\zeta_{p}^{\sigma}=\zeta_{p}^{-1}$ and thus

$(Y\zeta_{p}^{e})^{\sigma}=Y(\pm\zeta_{p}^{c}\zeta_{r}^{d})\zeta_{p}^{-e}=Y(\pm\zeta_{p}^{c-e}\zeta_{r}^{d})=Y(\pm\zeta_{p}^{e}\zeta_{r}^{d}).$

Let $Y_{1}=Y\zeta_{p}^{e}$. Then

$Y_{1}^{\sigma}=Y_{1}(\pm\zeta_{r}^{d}).$ (27)

Write $Y_{1}=\sum_{i=0}^{p-1}a_{i}\zeta_{p}^{i}$ with
$a_{i}\in\mathbb{Z}[\zeta_{r}]$. Note ${\cal M}(Y_{1})={\cal M}(Y)=n$. By
Result 7.1, we have

$(p-1)n=(p-1){\cal M}(Y_{1})=\sum_{i<j}^{p-1}{\cal M}(a_{i}-a_{j}).$ (28)

Let $t$ be the maximum number such that there are distinct indices
$i_{1},\ldots,i_{t}$ with $a_{i_{1}}=\cdots=a_{i_{t}}$. If $t\leq p/2$, then,
by (26), the right hand side of (28) is at least $p^{2}/4$ and thus $4n>p$,
contradicting the assumption $p>n^{2}+n+1$. Hence $t\geq p/2$. Note that, by
(26), the right hand side of (28) is at least $(p-t)t$. Hence $(p-t)t\leq
n(p-1)$. Note that $(p-t)t$ is decreasing for $t\in[p/2,p]$ and that
$(p-(p-n-1))(p-n-1)=n(p-1)+p-n-1-n^{2}>n(p-1)$, as $p>n^{2}+n+1$. Hence $t\geq
p-n$. Recall that $a_{i_{1}}=\cdots=a_{i_{t}}$. Note
$Y_{1}=\sum_{i=0}^{p-1}(a_{i}-a_{i_{1}})\zeta_{p}^{i}$. Thus, writing
$b_{i}=a_{i}-a_{i_{1}}$, we have $Y_{1}=\sum_{i=0}^{p-1}b_{i}\zeta_{p}^{i}$
and $|\\{i:b_{i}\neq 0\\}|\leq n$. If $|\\{i:b_{i}\neq 0\\}|=1$, then the
assertion of Theorem 7.2 holds. Hence, to complete the proof, it suffices to
show that

$2\leq|\\{i:b_{i}\neq 0\\}|\leq n$ (29)

leads to a contradiction. Define $X=\sum b_{i}g^{i}$ where $g$ is a generator
of $C_{p}$, the cyclic group of order $p$. Let $K$ be the kernel of the ring
homomorphism $\rho:\mathbb{Z}[\zeta_{r}][C_{p}]\to\mathbb{Z}[\zeta_{pr}]$
determined by $g\mapsto\zeta_{p}$. It is well known and straightforward to
verify that $K=\\{AC_{p}:A\in\mathbb{Z}[\zeta_{r}][C_{p}]\\}$. Note that
$\rho(X)=Y_{1}$, $\rho(X^{(-1)})=\overline{Y_{1}}$ and thus
$\rho(XX^{(-1)})=Y_{1}\overline{Y_{1}}=n$. Hence

$XX^{(-1)}=n+AC_{p}$ (30)

for some $A\in\mathbb{Z}[\zeta_{r}][C_{p}]$. Suppose $A\neq 0$. Then there are
at least $p-1$ nonzero coefficients on the right hand side of (30). On the
other hand, by (29), there are at most $n(n-1)$ nonzero coefficients on the
left hand side of (30). Hence $p\leq n(n-1)+1$ which contradicts the
assumptions. Thus $A=0$ and hence

$XX^{(-1)}=n.$ (31)

Recall that $X=\sum b_{i}g^{i}$. By (29), we can write
$X=\sum_{j=1}^{z}b_{i_{j}}g^{i_{j}}$ with $2\leq z\leq n$ and $b_{i_{j}}\neq
0$ for all $j$. Note

$XX^{(-1)}=\sum_{k=0}^{p-1}\left(\sum_{i_{r}-i_{s}\equiv k\bmod
p}b_{i_{r}}\overline{b_{i_{s}}}\right)g^{k}.$ (32)

Write $S=\\{i_{1},\ldots,i_{z}\\}$ and view $S$ as a subset of
$\mathbb{F}_{p}$. Note $2\leq|S|\leq n$. Suppose $S$ has a unique difference,
say, $k=i_{r}-i_{s}\in\mathbb{F}_{p}$ is a unique difference in $S$. Note that
$k\neq 0$, as $|S|\geq 2$ and thus $0$ is not a unique difference in $S$. In
view of (32), the coefficient of $g^{k}$ in $XX^{(-1)}$ is nonzero. But this
contradicts $XX^{(-1)}=n$. Thus $S$ has no unique difference.

Using (27), we get

$Y_{1}^{\sigma}=\sum_{i=0}^{p-1}b_{i}^{\sigma}\zeta_{p}^{-i}=\pm\zeta_{r}^{d}Y_{1}=\pm\zeta_{r}^{d}\sum_{i=0}^{p-1}b_{i}\zeta_{p}^{i}$

Hence, for $i\neq 0$, we have $b_{i}\neq 0$ if and only if $b_{p-i}\neq 0$.
This implies that $S$ is symmetric. As $S$ has no unique difference, we have
$p\leq 3^{|S|/2}$ by Theorem 1.2. As $|S|\leq n$, we conclude $p\leq 3^{n/2}$,
contradicting the assumptions. ∎

Example Let $p=107$, $q=2$, and $n=8$ in Theorem 7.2. Note that ${\rm
ord}_{p}(2)$ is even and that $p>\max\\{3^{n/2},n^{2}+n+1\\}$. Thus, for any
prime odd prime $r$ and $Y\in\mathbb{Z}[\zeta_{107r}]$ with $|Y|^{2}=8$, we
have $Y\eta\in\mathbb{Z}[\zeta_{r}]$ for some root of unity $\eta$.

Acknowledgement We are grateful to the referees for their careful reading of
the paper and suggestions which improved the exposition of the paper.

## References

  * [1] J. Browkin, B. Divis, and A. Schinzel: Addition of sequences in general fields, Monatsh. Math. 82 (1976), 261–268.
  * [2] E. Croot, T. Schoen: On sumsets and spectral gaps. Acta Arith.  136 (2009), 47–55.
  * [3] J. W. S. Cassels: On a conjecture of R. M. Robinson about sums of roots of unity. J. Reine Angew. Math.  238 (1969), 112–131.
  * [4] K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics 84, Springer 1990.
  * [5] K. H. Leung, B. Schmidt: Unique Sums and Differences in Finite Abelian Groups. Submitted.
  * [6] V. F. Lev: The rectifiability threshold in abelian groups. Combinatorica 28 (2008), 491–497.
  * [7] J. H. Loxton: On two problems of E. M. Robinson about sums of roots of unity. Acta Arith.  26 (1974), 159–174.
  * [8] Z. Nedev: An algorithm for finding a nearly minimal balanced set in ${\mathbb{F}}_{p}$. Math. Comp.  268 (2009), 2259–2267.
  * [9] Z. Nedev: Lower bound for balanced sets. Theoret. Comput. Sci.  460 (2012), 89–93.
  * [10] Z. Nedev, A. Quas: Balanced sets and the vector game. Int. J. Number Theory 4 (2008), 339–347.
  * [11] C. Norman: Finitely Generated Abelian Groups and Similarity of Matrices over a Field. Springer 2012.
  * [12] B. Schmidt: Cyclotomic integers and finite geometry. J. Am. Math. Soc. 12 (1999), 929–952.
  * [13] E. G. Straus: Differences of residues (mod $p$). J. Number Th. 8 (1976), 40–42.


# On scattering and damping of Toroidal Alfvén eigenmode by drift wave
turbulence

Liu Chen1,2,3, Zhiyong Qiu1,3, and Fulvio Zonca3,1 1Institute for Fusion
Theory and Simulation, School of Physics, Zhejiang University, Hangzhou, P.R.C
2Department of Physics and Astronomy, University of California, Irvine CA
92697-4575, U.S.A.
3 Center for Nonlinear Plasma Science and C.R. ENEA Frascati, C.P. 65, 00044
Frascati, Italy

###### Abstract

We demonstrate analytically that, in toroidal plasmas, scattering by drift
wave turbulence could lead to appreciable damping of toroidal Alfvén
eigenmodes via generation of short-wavelength electron Landau damped kinetic
Alfvén waves. A corresponding analytic expression of the damping rate is
derived, and found to be, typically, comparable to the linear drive by
energetic particles. The implications of this novel mechanism on the transport
and heating processes in burning plasmas are also discussed.

Microscopic drift wave (DW) turbulences are intrinsic to magnetically confined
plasmas such as tokamaks Horton (1999). These electrostatic waves have,
typically, frequencies in the range of electron/ion diamagnetic drift
frequencies and perpendicular wavelengths comparable to ion Larmor radii. On
the other hand, shear Alfvén waves (SAWs) or, more precisely, Alfvén
eigenmodes (AEs) excited by energetic particles (EPs) Cheng et al. (1985);
Fasoli et al. (2007); Chen and Zonca (2016), have been observed in current
experiments Wong et al. (1991); Heidbrink et al. (1993); Sharapov et al.
(2013), and predicted for future burning plasma experiments such as ITER
Tomabechi et al. (1991). Since these EP-driven AEs have direct bearing on the
confinement of EPs (including fusion $\alpha$ particles) and, consequently,
the plasma performance, it is crucial that the stability properties be
accurately assessed. In this Letter, we demonstrate analytically, for the
first time, that ambient DWs could lead to appreciable damping of toroidal
Alfvén eigenmode (TAE) Cheng et al. (1985) by cross-scale couplings via direct
nonlinear wave-wave interactions. Physically, this damping occurs because DWs
scatter the TAE into short-wavelength kinetic Alfvén waves (KAWs) Hasegawa and
Chen (1976) which are then Landau damped by, mainly, electrons. An explicit
analytic expression of the DWs-induced damping rate is derived and, for
typical tokamak parameters, is found to be comparable to the EP-driven TAE
growth rate. We also note that the novel nonlinear wave-wave interactions
carry significant implications to confinement and heating processes in burning
plasmas.

Let’s consider a tokamak plasma with circular magnetic surfaces and a large
aspect ratio, i.e., $\epsilon\equiv r/R<1$. Here, $r$ and $R$ are,
respectively, the minor and major radii of the torus. We, furthermore, take
$\beta\equiv P/B^{2}\sim O(\epsilon^{2})\ll 1$, the equilibrium distributions
be Maxwellian, and the density be nonuniform. However, in order, as a paradigm
model, to simplify the analysis, we have, for now, neglected the effects of
finite temperature gradients and trapped particles. Thus, the present work is
focused on the roles played by electron drift waves in scattering and damping
of TAE. With $\beta\ll 1$, magnetic compression can be neglected, and we may
adopt $\delta\phi$ and $\delta A_{\parallel}$ as the field variables. Here,
$\delta\phi$ is the scalar potential and $\delta A_{\parallel}$ is the
parallel component of the vector potential; i.e.,
$\delta\mathbf{A}\simeq\delta A_{\parallel}\mathbf{b}$ and
$\mathbf{b}\equiv\mathbf{B}_{0}/B_{0}$. The perturbed distribution function,
$\delta f_{j}$ with $j=e,i$, then obeys the following nonlinear gyrokinetic
equations Frieman and Chen (1982)

$\displaystyle\delta f_{j}=-(e/T)_{j}\delta\phi
F_{Mj}+\exp(-\bm{\rho}_{j}\cdot\nabla)\delta g_{j},$ (1)

and

$\displaystyle\left(\partial_{t}+v_{\parallel}\mathbf{b}\cdot\nabla+\mathbf{v}_{d}\cdot\nabla+\langle\delta\mathbf{u}_{gj}\rangle_{\alpha}\cdot\nabla\right)\delta
g_{j}$ (2) $\displaystyle=$
$\displaystyle(e/T)_{j}F_{Mj}(\partial_{t}+i\omega_{*j})\langle\exp(\bm{\rho}_{j}\cdot\nabla)\delta
L\rangle_{\alpha}.$

Here, $F_{Mj}$ is the Maxwellian distribution, $\Omega_{j}=(eB_{0}/mc)_{j}$,
$\mathbf{v}_{d}=\mathbf{b}\times[(v^{2}_{\perp}/2)\nabla\ln
B_{0}+v^{2}_{\parallel}\mathbf{b}\cdot\nabla\mathbf{b}]$ is the magnetic drift
velocity, $\langle A\rangle_{\alpha}$ denotes the gyro-phase averaging of $A$,
$\langle\delta\mathbf{u}_{gj}\rangle_{\alpha}=(c/B_{0})\mathbf{b}\times\nabla\langle\exp(\bm{\rho}_{j}\cdot\nabla)\delta
L\rangle_{\alpha}$, $\delta L=\delta\phi-v_{\parallel}\delta A_{\parallel}/c$,
and $\omega_{*j}=-i(cT/eB_{0})_{j}\mathbf{b}\times\nabla\ln N_{j}\cdot\nabla$.
The governing field equations, meanwhile, are the quasi-neutrality condition

$\displaystyle\sum_{j=e,i}[(N_{0}e^{2}/T)_{j}\delta\phi-
e_{j}\langle(J_{k}\delta g)_{j}\rangle_{v}]=0,$ (3)

and, for Alfvén waves, the nonlinear gyrokinetic vorticity equation in the
wave-vector form Chen et al. (2001)

$\displaystyle ik_{\parallel}\delta J_{\parallel
k}+(N_{0}e^{2}/T)_{j}(1-\Gamma_{k})(\partial_{t}+i\omega_{*i})_{k}\delta\phi_{k}$
$\displaystyle-\sum_{j}\langle e_{j}J_{k}\omega_{d}\delta
g_{j}\rangle=\sum_{\mathbf{k}=\mathbf{k}^{\prime}+\mathbf{k}^{\prime\prime}}\Lambda^{k^{\prime}}_{k^{\prime\prime}}\left\\{\delta
A_{\parallel k^{\prime}}\delta J_{\parallel k^{\prime\prime}}/c\right.$
$\displaystyle\hskip
50.00008pt\left.-e_{i}\langle(J_{k}J_{k^{\prime}}-J_{k^{\prime\prime}})\delta
L_{k^{\prime}}\delta g_{k^{\prime\prime}i}\rangle_{v}\right\\}.$ (4)

We note that, in equations (3) and (4), $\mathbf{k}$ should be strictly
understood as an operator; i.e., $\mathbf{k}=-i\nabla$,
$ik_{\parallel}=\mathbf{b}\cdot\nabla$, and
$k^{2}_{\perp}=-\nabla^{2}_{\perp}$, $\langle\cdots\rangle_{v}=\int
d^{3}\mathbf{v}(\cdots)$, $J_{k}=J_{0}(k_{\perp}\rho)$,
$\Gamma_{k}=I_{0}(b_{k})\exp(-b_{k})$, $b_{k}=k^{2}_{\perp}\rho^{2}_{i}$,
$\rho_{i}=v_{ti}/\Omega_{i}$ with $v_{ti}=\sqrt{T_{i}/m_{i}}$, $I_{0}$ being
the modified Bessel function, and $J_{0}(k_{\perp}\rho_{e})\simeq 1$ since
$|k_{\perp}\rho_{e}|^{2}\ll 1$. In equation (4), $\delta J_{\parallel}$
satisfies the Ampére’s law $\delta
J_{\parallel}=-(c/4\pi)\nabla^{2}_{\perp}\delta A_{\parallel}$ and
$\omega_{d}=\mathbf{v}_{d}\cdot\mathbf{k}_{\perp}$. Meanwhile, on the right
hand side of equation (4),
$\Lambda^{k^{\prime}}_{k^{\prime\prime}}=(c/B_{0})\mathbf{b}\cdot(\mathbf{k}^{\prime\prime}\times\mathbf{k}^{\prime})$,
and the two nonlinear terms correspond, respectively, to the Maxwell and
generalized gyrokinetic ion Reynolds stresses. Note also that, since the
ambient DWs are electrostatic, the Maxwell stress does not contribute to the
scattering of TAE and, hence, can be ignored in the present analysis.

Adopting the ballooning-mode representation Connor et al. (1978), we let the
ambient stationary DW fluctuation be
$\delta\phi_{DW}=\sum_{n_{s}}(\delta\phi_{s}+c.c.)/2$, and

$\displaystyle\delta\phi_{s}$ $\displaystyle=$
$\displaystyle(T_{e}/e)A_{n_{s}}\exp(-i\omega_{n_{s}}t+in_{s}\xi)$ (5)
$\displaystyle\times\sum_{m_{s}}\exp(-im_{s}\theta)\Phi_{s}(n_{s}q-m_{s}),$

where $\xi$ and $\theta$ are, respectively, the toroidal and poloidal angle
coordinates, $q(r)$ is the safety factor, and
$\int^{\infty}_{-\infty}|\Phi_{s}|^{2}d(n_{s}q-m_{s})=1$ is the normalization.
We now let $\Omega_{0}=(\omega_{0},n_{0})$ be the test TAE with frequency
$\omega_{0}$ and toroidal mode number $n_{0}$. The scattering would occur for
each $\delta\phi_{s}$, denoted as $\Omega_{s}=(\omega_{n_{s}},n_{s})$, and
generate upper and lower sidebands denoted, respectively, as
$\Omega_{+}=(\omega_{+}=\omega_{0}+\omega_{n_{s}},n_{+}=n_{0}+n_{s})$ and
$\Omega_{-}=(\omega_{-}=\omega_{n_{s}}-\omega_{0},n_{-}=n_{s}-n_{0})$. As will
be shown later in the detailed analysis, the nonlinearly generated
$\Omega_{\pm}$ quasimodes correspond to mode-converted high-$n$ KAWs Hasegawa
and Chen (1976); which are Landau damped by, mainly, electrons and, as a
consequence, the scattering will contribute to the damping of the test TAE.
The net damping rate is then given by the sum of all $\Omega_{s}$ over
$n_{s}$.

First, let us investigate the $\Omega_{+}$ channel. Analysis for the
$\Omega_{-}$ is similar. Noting that
$|k_{\parallel}v_{te}|\gg|\omega_{k}|\gg|k_{\parallel}v_{ti}|$ for all the
modes considered here, we then find, from equation (2), $\delta g_{k}=\delta
g^{(1)}_{k}+\delta g^{(2)}_{k}$, where

$\displaystyle\delta
g^{(1)}_{+i}\simeq\left(\frac{e}{T_{i}}\right)F_{Mi}\left(1-\frac{\omega_{*i}}{\omega}\right)_{+}J_{+}\delta\phi_{+},$
(6)

and

$\displaystyle\delta
g^{(1)}_{+e}\simeq-\left(\frac{e}{T_{e}}\right)F_{Me}\left(1-\frac{\omega_{*e}}{\omega}\right)_{+}\delta\psi_{+},$
(7)

are the linear responses, and $\delta\psi_{k}=(\omega\delta
A_{\parallel}/ck_{\parallel})_{k}$ is the effective potential due to the
$\partial_{t}\delta A_{\parallel}/c$ induced parallel electric field. The
nonlinear responses, $\delta g^{(2)}_{k}$, meanwhile, are given by

$\displaystyle\delta
g^{(2)}_{+i}\simeq-i(\Lambda^{s}_{0}/2\omega_{0})J_{0}J_{s}\frac{e}{T_{i}}F_{Mi}\left(\frac{\omega_{*i}}{\omega}\right)_{s}\delta\phi_{s}\delta\phi_{0},$
(8)

with
$\Lambda^{s}_{0}=(c/B_{0})\mathbf{b}\cdot(\mathbf{k}_{0\perp}\times\mathbf{k}_{s\perp})$,
and, noting $\Omega_{s}$ being an electrostatic mode, $\delta
g^{(2)}_{+e}\simeq 0$. Applying the quasi-neutrality condition, equation (3),
we then obtain

$\displaystyle\delta\psi_{+}=\sigma_{*+}\delta\phi_{+}+i(\Lambda^{s}_{0}/2\omega_{0})D_{+}\delta\phi_{0}\delta\phi_{s},$
(9)

where

$\displaystyle\sigma_{*k}$ $\displaystyle=$
$\displaystyle[1+\tau-\tau\Gamma_{k}(1-\omega_{*i}/\omega)_{k}]/(1-\omega_{*e}/\omega)_{k},$
(10) $\displaystyle D_{+}$ $\displaystyle=$
$\displaystyle\tau(\omega_{*i}/\omega)_{s}F_{+}/(1-\omega_{*e}/\omega)_{+},$
(11)

$\tau=T_{e}/T_{i}$, and $F_{+}=\langle
J_{+}J_{0}J_{s}F_{Mi}/N_{0}\rangle_{v}$. Applying the nonlinear gyrokinetic
vorticity equation, equation (4), meanwhile, yields, after straightforward
algebra,

$\displaystyle\tau
b_{+}\left[\left(1-\frac{\omega_{*i}}{\omega}\right)_{+}\frac{(1-\Gamma_{+})}{b_{+}}\delta\phi_{+}-\left(\frac{V^{2}_{A}}{b}\frac{k_{\parallel}bk_{\parallel}}{\omega^{2}}\right)_{+}\delta\psi_{+}\right]$
(12) $\displaystyle=$
$\displaystyle-i(\Lambda^{s}_{0}/2\omega_{0})\gamma_{+}\delta\phi_{s}\delta\phi_{0},$

where $V_{A}=B_{0}/\sqrt{4\pi N_{0}m_{i}}$ is the Alfvén speed, and

$\displaystyle\gamma_{+}=\tau[\Gamma_{s}-\Gamma_{0}+(\omega_{*i}/\omega)_{s}(F_{+}-\Gamma_{s})].$
(13)

Combining equations (9) and (12) then leads to the desired equation
demonstrating the nonlinear generation of $\Omega_{+}$ quasimode by
$\Omega_{s}$ and $\Omega_{0}$;

$\displaystyle\tau
b_{+}\epsilon_{A+}\delta\phi_{+}=-i(\Lambda^{s}_{0}/2\omega_{0})\beta_{+}\delta\phi_{s}\delta\phi_{0},$
(14)

where

$\displaystyle\epsilon_{Ak}=\left(1-\frac{\omega_{*i}}{\omega}\right)_{k}\frac{(1-\Gamma_{k})}{b_{k}}-\left(\frac{V^{2}_{A}}{b}\frac{k_{\parallel}bk_{\parallel}}{\omega^{2}}\right)_{k}\sigma_{*k}$
(15)

is the linear SAW/KAW operator, and

$\displaystyle\beta_{+}$ $\displaystyle=$
$\displaystyle\tau(\Gamma_{s}-\Gamma_{0})+\tau\left(\frac{\omega_{*i}}{\omega}\right)_{s}\left[F_{+}-\Gamma_{s}-\right.$
(16) $\displaystyle+$
$\displaystyle\left.\left(\frac{V^{2}_{A}}{b}\frac{k_{\parallel}bk_{\parallel}}{\omega^{2}}\right)_{+}\frac{\tau
b_{+}F_{+}}{(1-\omega_{*e}/\omega)_{+}}\right].$

Next, we consider the feeding back to $\Omega_{0}$ via nonlinear coupling
between $\Omega_{+}$ and $\Omega^{*}_{s}$. Note that the test TAE $\Omega_{0}$
evolution is also affected, on the same footing, by the other channel of
$\Omega_{s}$ and $\Omega^{*}_{-}$ coupling, and the treatment is similar to
the analysis of $\Omega_{+}$ and $\Omega^{*}_{s}$ couplings presented in the
following equations (17) to (28). The $\Omega_{-}$ contribution will be added
in equation (29). Noting that $\delta g_{+i}=\delta g^{(1)}_{+i}+\delta
g^{(2)}_{+i}$ given by equations (6) and (8), we readily find

$\displaystyle\delta g^{(2)}_{0i}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{e}{T}\right)_{i}F_{Mi}\left(\frac{\omega_{*i}}{\omega}\right)_{s}\left[i(\Lambda^{s}_{0}/2\omega_{0})J_{s}J_{+}\delta\phi^{*}_{s}\delta\phi_{+}\right.$
(17)
$\displaystyle\left.+(\Lambda^{s}_{0}/2\omega_{0})^{2}J_{0}J^{2}_{s}|\delta\phi_{s}|^{2}\delta\phi_{0}\right].$

The second nonlinear term in equation (17) is due to $\delta g^{(2)}_{+i}$,
and may be regarded as the diagonal nonlinear term. The quasi-neutrality
condition then yields

$\displaystyle\delta\psi_{0}=\left(\sigma_{*0}+\alpha_{0}|\delta\phi_{s}|^{2}\right)\delta\phi_{0}-i(\Lambda^{s}_{0}/2\omega_{0})D^{+}_{0}\delta\phi^{*}_{s}\delta\phi_{+},$
(18)

where

$\displaystyle
D^{+}_{0}=\tau(\omega_{*i}/\omega)_{s}F_{+}/(1-\omega_{*e}/\omega)_{0},$ (19)

and
$\alpha_{0}=-(\Lambda^{s}_{0}/2\omega_{0})^{2}\tau(\omega_{*i}/\omega)_{s}F_{2}$
with $F_{2}\equiv\langle J^{2}_{0}J^{2}_{s}F_{Mi}/N_{0}\rangle_{v}$ leads to
negligible nonlinear frequency shift. The nonlinear gyrokinetic vorticity
equation, equation (4), meanwhile, yields,

$\displaystyle\tau
b_{0}\left\\{\left[\left(1-\frac{\omega_{*i}}{\omega}\right)_{0}\frac{(1-\Gamma_{0})}{b_{0}}+\alpha^{+}_{0}|\delta\phi_{s}|^{2}\right]\delta\phi_{0}\right.$
$\displaystyle\left.-\left(\frac{V^{2}_{A}}{b}\frac{k_{\parallel}bk_{\parallel}}{\omega^{2}}\right)_{0}\delta\psi_{0}\right\\}=i\left(\frac{\Lambda^{s}_{0}}{2\omega_{0}}\right)\gamma^{+}_{0}\delta\psi^{*}_{s}\delta\phi_{+},$
(20)

where

$\displaystyle\gamma^{+}_{0}=\tau\left[\Gamma_{s}-\Gamma_{+}+(\omega_{*i}/\omega)_{s}(F_{+}-F_{s})\right],$
(21)

and
$\alpha^{+}_{0}=-(\Lambda^{s}_{0}/2\omega_{0})^{2}(\omega_{*i}/\omega)_{s}(F_{2}-F_{+})/b_{0}$
also leads to negligible nonlinear frequency shift. Combining equations (18)
and (20) and neglecting the nonlinear frequency shift, we obtain the desired
equation

$\displaystyle\tau
b_{0}\epsilon_{A0}\delta\phi_{0}=i(\Lambda^{s}_{0}/2\omega_{0})\beta^{+}_{0}\delta\phi^{*}_{s}\delta\phi_{+},$
(22)

where

$\displaystyle\beta^{+}_{0}\simeq(\sigma_{*0}\sigma_{*+}-\sigma_{s}F_{+}/\Gamma_{s})/\sigma_{*0}\equiv\hat{\beta}_{+}/\sigma_{*0}.$
(23)

In deriving equation (23), we have noted, in the nonlinear analysis, that
$\Omega_{s}$ and $\Omega_{0}$ are normal modes; such that
$\tau(\omega_{*i}/\omega)_{s}\simeq-(1+\tau-\tau\Gamma_{s})/\Gamma_{s}$ and
$|\epsilon_{A0}\delta\phi_{0}|\simeq O(|\delta\phi_{s}|^{2})|\delta\phi_{0}|$.

Equations (14) and (22) are the coupled equations for $\delta\phi_{0}$ and
$\delta\phi_{+}$; which yield, formally,

$\displaystyle\tau
b_{0}\epsilon_{A0}\delta\phi_{0}=\left[\left(\frac{\Lambda^{s}_{0}}{2\omega_{0}}\right)^{2}\beta^{+}_{0}\delta\phi^{*}_{s}\frac{\beta_{+}}{\left(\tau
b_{+}\epsilon_{A+}\right)}\delta\phi_{s}\right]\delta\phi_{0}.$ (24)

Equation (24) can be solved analytically by employing the scale separation
between $\delta\phi_{0}$ and $\delta\phi_{s}$. More specifically, as TAE is
typically excited by EPs, one has
$k_{0\theta}\rho_{EP}=(n_{0}q/r)\rho_{EP}\sim O(1)$ and, for DWs,
$k_{s\theta}\rho_{i}\sim O(1)$; and, hence, $|n_{0}/n_{s}|\sim
O(\rho_{i}/\rho_{EP})\sim O(|T_{i}/T_{EP}|^{1/2})\ll 1$. Denoting
$\mathbf{x}_{s}=(R/n_{s},r/m_{s},1/n_{s}q^{\prime})$ and
$\mathbf{x}_{0}=(R/n_{0},r/m_{0},1/n_{0}q^{\prime})$ as, respectively, the
microscopic DW and macro/mesoscopic TAE scales, and expanding
$\delta\phi_{0}=\Phi_{0}(\mathbf{x}_{0})+\tilde{\Phi}_{0}(\mathbf{x}_{s},\mathbf{x}_{0})$
with $|\tilde{\Phi}_{0}|/|\Phi_{0}|\sim O(|\delta\phi_{s}|^{2})\ll 1$,
equation (24) then becomes, after averaging over the $\mathbf{x}_{s}$ scale,

$\displaystyle\tau
b_{0}\epsilon_{A0}\Phi_{0}=\left\langle\left(\frac{\Lambda^{s}_{0}}{2\omega_{0}}\right)^{2}\beta^{+}_{0}\delta\phi^{*}_{s}\frac{\beta_{+}}{\tau
b_{+}\epsilon_{A+}}\delta\phi_{s}\right\rangle_{s}\Phi_{0},$ (25)

where noting equation (5) and denoting $z_{s}=n_{s}q-m_{s}$ as the
dimensionless microscopic DW radial coordinate,

$\displaystyle\langle(\cdots)|\delta\phi_{s}|^{2}\rangle_{s}$ (26)
$\displaystyle=$
$\displaystyle\left(\frac{T_{e}}{e}\right)^{2}|A_{n_{s}}|^{2}\int^{1/2}_{-1/2}dz_{s}(\cdots)\sum_{m_{s}}|\Phi_{s}(z_{s})|^{2}$
$\displaystyle=$
$\displaystyle\left(\frac{T_{e}}{e}\right)^{2}|A_{n_{s}}|^{2}\int^{\infty}_{-\infty}dz_{s}(\cdots)|\Phi_{s}(z_{s})|^{2}.$

Focusing, furthermore, on the stability due to $\mbox{Im}(1/\epsilon_{A+})$
which can be formally expressed as
$\mbox{Im}(1/\epsilon_{A+})=-\pi\delta(\epsilon_{A+})$. Noting also that, in
equation (25),
$\beta_{+}\delta(\epsilon_{A+})=(\hat{\beta}_{+}/\sigma_{*+})\delta(\epsilon_{A+})$
and $\hat{\beta}_{+}$ given by equation (23). Applying the two-scale
expansion; i.e., $k^{2}_{+\perp}\simeq
k^{2}_{s\perp}+2\mathbf{k}_{s\perp}\cdot\mathbf{k}_{0\perp}\simeq
k^{2}_{s\perp}+2k_{sr}k_{0r}$, to $\hat{\beta}_{+}$, we obtain

$\displaystyle\hat{\beta}_{+}\delta\phi_{s}\simeq\frac{T_{e}}{e}A_{n_{s}}e^{i(n_{s}\xi-\omega_{n_{s}}t)}\sum_{m_{s}}e^{-im_{s}\theta}\left(\tau+\frac{\sigma_{s}}{2\Gamma_{s}}\right)$
$\displaystyle\times(\partial\Gamma_{s}/\partial
b_{s})(2ik_{0r}\rho_{i})(k_{s\theta}\rho_{i})\hat{s}\partial\Phi_{s}/\partial
z_{s},$ (27)

and $\hat{s}=rq^{\prime}/q$. Meanwhile, from equation (23),
$\beta^{+}_{0}\delta\phi^{*}_{s}=[\hat{\beta}_{+}\delta\phi_{s}]^{*}/\sigma_{*0}$.
We remark that, $\mbox{Im}(1/\epsilon_{A+})=-\pi\delta(\epsilon_{A+})$
physically corresponds to absorption of the mode-converted $\Omega_{+}$ KAW
via electron Landau damping and, since the absorption occurs predominantly in
a narrow region near the SAW resonance layer Hasegawa and Chen (1976), its
absorption rate can be quantitatively estimated by the Aflvén resonance
absorption Chen and Hasegawa (1974); Zonca and Chen (1993, 1996);

$\displaystyle\mbox{Im}(1/\epsilon_{A+})=-\pi\delta(\epsilon_{A+})\simeq-(\pi/4\sigma_{*+})\delta(z^{2}_{s}-z^{2}_{+}).$
(28)

Here, we have noted $\epsilon_{A+}$ given by equation (15),
$k_{+\parallel}\simeq k_{s\parallel}=(n_{s}q-m_{s})/qR=z_{s}/qR$ and
$z^{2}_{+}=(1-\omega_{*i}/\omega)_{+}(1-\Gamma_{+})(\omega/\omega_{A})^{2}_{+}/(b_{+}\sigma_{*+})$,
$\omega_{A}=V_{A}/qR$, $(\omega/\omega_{A})_{+}\simeq 1/2$, and $b_{+}\simeq
b_{s}\simeq b_{s\theta}$ for typical moderately to strongly ballooning DWs.
Substituting equations (27) and (28) into equation (25), and summing over all
$\Omega_{s}$ over $n_{s}$, the TAE equation then becomes

$\displaystyle\tau
b_{0}\left[\epsilon_{A0}+i\nu(k_{0r}\rho_{i})^{2}\right]\Phi_{0}=0,$ (29)

where $\nu=\nu_{+}+\nu_{-}$, and

$\displaystyle\nu_{\pm}$ $\displaystyle\simeq$
$\displaystyle\pi\left(\frac{\Omega_{ci}}{\omega_{0}}\right)^{2}\sum_{n_{s}}|A_{n_{s}}|^{2}\left[\left(\tau+\frac{\sigma_{s}}{2\Gamma_{s}}\right)\frac{\partial\Gamma_{s}}{\partial
b_{s\theta}}\right]^{2}$ (30) $\displaystyle\times
b_{s\theta}\hat{s}^{2}\left(\sigma^{2}_{s\pm}z_{\pm}\right)^{-1}|\partial\Phi_{s}/\partial
z_{s}|^{2}_{z_{\pm}}.$

In deriving equation (29), we have noted, again, that while the $\Omega_{+}$
channel results in the $\nu_{+}$ damping, the analysis of the $\Omega_{-}$
channel is similar and yields the $\nu_{-}$ term.

Equation (29) can be readily solved perturbatively in the ballooning space,
$\eta$. Let $\hat{\Phi}(\eta)$ be the lowest-order eigenmode; i.e.,
$\hat{b}_{0}\hat{\epsilon}_{A0}(\eta,\partial_{\eta},\omega_{0})\hat{\Phi}_{0}(\eta)=0$
and $\omega_{0}=\omega_{0r}+i\gamma_{AD}$, with $\gamma_{AD}$ being the TAE
damping rate induced by DW scattering. Here, from equation (15),

$\displaystyle\hat{b}_{0}\hat{\epsilon}_{A0}=\hat{b}_{0}-\left(\frac{\omega_{A}}{\omega_{0}}\right)^{2}\partial_{\eta}\hat{b}_{0}(1+2\epsilon_{0}\cos\eta)\partial_{\eta},$

$\hat{b}_{0}=(1+\hat{s}^{2}\eta^{2})$, $\epsilon_{0}=2(\Delta^{\prime}+r/R)$
and $\Delta^{\prime}\sim r/R$ is the radial derivative of the Shafranov shift.
Equation (29) then gives

$\displaystyle\frac{2\gamma_{AD}}{\omega_{0r}}\left\langle\hat{\Phi}_{0}\hat{b}_{0}\hat{\Phi}_{0}\right\rangle_{\eta}=-\left\langle\hat{\Phi}_{0}\hat{b}_{0}\nu
b_{0\theta}\hat{s}^{2}\eta^{2}\hat{\Phi}_{0}\right\rangle_{\eta}.$ (31)

Here, $\langle A\rangle_{\eta}=\int^{\infty}_{-\infty}Ad\eta$. Given that, for
TAE Cheng et al. (1985),
$\hat{\Phi}_{\eta}=[A\cos(\eta/2)+B\sin(\eta/2)]\exp(-\lambda|\eta|)/\hat{b}^{1/2}_{0}$,
and
$\lambda^{2}=[(1+\epsilon_{0})/4-(\omega_{0}/\omega_{A})^{2}][(\omega_{0}/\omega_{A})^{2}-(1-\epsilon_{0})/4]\sim
O(\epsilon^{2})$; equation (31) then yields

$\displaystyle\frac{\gamma_{AD}}{\omega_{0r}}=-\frac{1}{4}\frac{\nu
b_{0\theta}\hat{s}^{2}}{\lambda^{2}}.$ (32)

Equation (32) is the desired analytic expression for TAE damping rate due to
scattering by DWs. To obtain a quantitative estimate of $\gamma_{AD}$, we take
$|\Phi_{s}(z_{s})|=(\pi^{-1/4}\Delta^{-1/2}_{s})\exp(-z^{2}_{s}/2\Delta^{2}_{s})$
with $\Delta_{s}\gtrsim 1$ such that neighboring poloidal harmonics overlap to
produce the ballooning mode structure. We then have, in equation (30),
$|\partial\Phi_{s}/\partial
z_{s}|^{2}\simeq(z^{2}_{\pm}/\pi^{1/2}\Delta^{5}_{s})$; noting $|z_{\pm}|<1/2$
and $\Delta_{s}\gtrsim 1$. Taking typical parameters,
$|\Omega_{ci}/\omega_{0}|\sim O(10^{2})$,
$\sum_{n_{s}}|A_{n_{s}}|^{2}\simeq|e\delta\phi_{DW}/T_{e}|^{2}\simeq|\delta
n_{DW}/N_{0}|^{2}\sim O(10^{-4})$, $b_{s\theta}\sim\hat{s}\sim\tau\sim O(1)$,
$4\lambda^{2}\sim O(\epsilon^{2})\sim O(10^{-1}-10^{-2})$, and
$|z_{\pm}|/\Delta^{5}_{s}\sim O(10^{-1})$, we find
$(\gamma_{AD}/\omega_{0r})\simeq-O(1-10)b_{0\theta}$. Here, $\delta n_{DW}$ is
the DW induced density fluctuation. As the EP drive maximizes around
$k_{0\theta}\rho_{EP}\sim O(1)$, we have
$b_{0\theta}\simeq(k_{0\theta}\rho_{EP})^{2}(T_{i}/T_{EP})\simeq T_{i}/T_{EP}$
and for, typically, $T_{i}/T_{EP}\sim O(10^{-2})$,
$(\gamma_{AD}/\omega_{0r})\simeq-O(10^{-2}-10^{-1})$. We note that theoretical
studies have shown that TAE instabilities excited by EPs have typically growth
rates $\gamma_{EP}/\omega_{0r}\sim O(10^{-2})$ Chen and Zonca (2016),
$\gamma_{AD}$, thus, could be generally comparable to $\gamma_{EP}$, and,
consequently, could significantly reduce or even, with sufficiently large DW
intensity, suppress TAE fluctuations. This raises the interesting implication
that, in a burning plasma, with the presence of microscopic DW turbulence, the
critical gradient of $\alpha$-particles could be upshifted; leading to
improved $\alpha$-particle confinement and, thereby, enhanced thermal plasma
heating. Another important implication is that, as the high-$n$ KAW quasi-
modes are dissipated, mainly, by electrons, the scattering and damping
processes discussed here could provide a collisionless channel of transferring
$\alpha$-particle energy to electrons via excitations of TAEs. Indeed,
assuming the dissipated TAE wave energy is absorbed by electrons, we readily
find that

$\displaystyle(d\beta_{e}/dt)_{AD}$ $\displaystyle=$ $\displaystyle
4|\gamma_{AD}||\delta B_{\perp}/B_{0}|^{2}$ (33) $\displaystyle\simeq$
$\displaystyle O(10^{-2}-10^{-1})4\omega_{0r}|\delta B_{\perp}/B_{0}|^{2}.$

Taking, typically, $|\delta B_{\perp}/B_{0}|\simeq 5\times 10^{-4}$ Heidbrink
et al. (2007) and $\omega_{0r}\simeq 10^{6}/s$, we have,
$(d\beta_{e}/dt)_{AD}\simeq O(10^{-2}-10^{-1}s^{-1})$. As a comparison,
electron heating by EPs ($\alpha$-particles) with, e.g., a slowing down time
$\tau_{SD}\sim O(10^{2}\mbox{ms})$, is $(d\beta_{e}/dt)_{EP}\simeq
O(10s^{-1})\beta_{EP}\simeq O(1-10s^{-1})\beta_{e}\simeq
O(10^{-2}-10^{-1}s^{-1})$. Here, we have assumed $\beta_{EP}\sim
O(\tau_{SD}/\tau_{E})\beta_{e}$ with $\tau_{E}$ being the electron energy
confinement time is, typically, a fraction of $\beta_{e}$ and, again,
$\beta_{e}\sim O(10^{-2})$. These estimates suggest that
$(d\beta_{e}/dt)_{AD}$ could, potentially, contribute significantly to
“anomalous” electron heating in burning plasmas.

In conclusion, we have employed the nonlinear gyrokinetic theory and
investigated analytically the effects of ambient stationary DW turbulence on
the linear stability of TAE via direct nonlinear wave-wave interactions. Our
analysis demonstrates that the scattering of TAE by DWs could lead to
appreciable damping of TAE due to electron Landau damping of the nonlinearly
generated high-$n$ KAW quasimodes. A corresponding analytical expression of
the damping rate is derived and, for typical parameters, the predicted damping
rate could be comparable to the TAE growth rate driven by EPs. Our result,
thus, suggests not only an additional TAE damping mechanism; but also carries
interesting implications to improved $\alpha$-particle confinement as well
enhanced collisionless heating of electrons in burning plasmas. Finally, we
remark that, as noted earlier, our analysis adopts a paradigm electron DW
model, where we keep nonuniform densities but assume uniform temperatures. It
is obviously desirable to extend the present analysis to include finite
temperature gradients and possible trapped-particle effects, as well as to
other types of AEs; e.g., reversed shear Alfvén eigenmode (RSAE) Sharapov et
al. (2001); Zonca et al. (2002) and beta-induced Alfvén eigenmode (BAE)
Heidbrink et al. (1993); Zonca et al. (1996). These and other possible
extensions are currently under investigation and will be reported in the
future.

This work is supported by the National Key Research and Development Program of
China under Grant No. 2017YFE0301900, National Science Foundation of China
under Grant Nos. 11235009 and 11875233, and “Users of Excellence program of
Hefei Science Center CAS under Contract No. 2021HSC-UE016”. This work has been
carried out within the framework of the EUROfusion Consortium, funded by the
European Union via the Euratom Research and Training Programme (Grant
Agreement No 101052200 - EUROfusion). Views and opinions expressed are however
those of the author(s) only and do not necessarily reflect those of the
European Union or the European Commission. Neither the European Union nor the
European Commission can be held responsible for them.

## References

  * Horton (1999) W. Horton, Rev. Mod. Phys. 71, 735 (1999).
  * Cheng et al. (1985) C. Cheng, L. Chen, and M. Chance, Ann. Phys. 161, 21 (1985).
  * Fasoli et al. (2007) A. Fasoli, C. Gormenzano, H. Berk, B. Breizman, S. Briguglio, D. Darrow, N. Gorelenkov, W. Heidbrink, A. Jaun, S. Konovalov, et al., Nuclear Fusion 47, S264 (2007).
  * Chen and Zonca (2016) L. Chen and F. Zonca, Review of Modern Physics 88, 015008 (2016).
  * Wong et al. (1991) K. L. Wong, R. J. Fonck, S. F. Paul, D. R. Roberts, E. D. Fredrickson, R. Nazikian, H. K. Park, M. Bell, N. L. Bretz, R. Budny, et al., Phys. Rev. Lett. 66, 1874 (1991).
  * Heidbrink et al. (1993) W. Heidbrink, E. Strait, M. Chu, and A. Turnbull, Phys. Rev. Lett. 71, 855 (1993).
  * Sharapov et al. (2013) S. Sharapov, B. Alper, H. Berk, D. Borba, B. Breizman, C. Challis, I. Classen, E. Edlund, J. Eriksson, A. Fasoli, et al., Nuclear Fusion 53, 104022 (2013).
  * Tomabechi et al. (1991) K. Tomabechi, J. Gilleland, Y. Sokolov, R. Toschi, and ITER Team, Nuclear Fusion 31, 1135 (1991).
  * Hasegawa and Chen (1976) A. Hasegawa and L. Chen, Physics of Fluids 19, 1924 (1976).
  * Frieman and Chen (1982) E. A. Frieman and L. Chen, Physics of Fluids 25, 502 (1982).
  * Chen et al. (2001) L. Chen, Z. Lin, R. B. White, and F. Zonca, Nuclear fusion 41, 747 (2001).
  * Connor et al. (1978) J. Connor, R. Hastie, and J. Taylor, Phys. Rev. Lett. 40, 396 (1978).
  * Chen and Hasegawa (1974) L. Chen and A. Hasegawa, The Physics of Fluids 17, 1399 (1974).
  * Zonca and Chen (1993) F. Zonca and L. Chen, Physics of Fluids B: Plasma Physics 5, 3668 (1993).
  * Zonca and Chen (1996) F. Zonca and L. Chen, Physics of Plasmas 3, 323 (1996).
  * Heidbrink et al. (2007) W. W. Heidbrink, N. N. Gorelenkov, Y. Luo, M. A. Van Zeeland, R. B. White, M. E. Austin, K. H. Burrell, G. J. Kramer, M. A. Makowski, G. R. McKee, et al. (the DIII-D team), Phys. Rev. Lett. 99, 245002 (2007).
  * Sharapov et al. (2001) S. Sharapov, D. Testa, B. Alper, D. Borba, A. Fasoli, N. Hawkes, R. Heeter, M. Mantsinen, and M. Von Hellermann, Physics Letters A 289, 127 (2001).
  * Zonca et al. (2002) F. Zonca, S. Briguglio, L. Chen, S. Dettrick, G. Fogaccia, D. Testa, and G. Vlad, Physics of Plasmas 9, 4939 (2002).
  * Zonca et al. (1996) F. Zonca, L. Chen, and R. A. Santoro, Plasma Physics and Controlled Fusion 38, 2011 (1996).