# Beyond canonical decoupling

Ciprian Demeter  Department of Mathematics, Indiana University, 831 East 3rd
St., Bloomington IN 47405<EMAIL_ADDRESS>
###### Abstract.

We introduce two families of inequalities. Large ensemble decoupling is
connected to the continuous restriction phenomenon. Tight decoupling is
connected to the discrete Restriction conjecture for the sphere. Our
investigation opens new grounds and answers some questions.

###### Key words and phrases:

exponential sums, decoupling, restriction conjecture

The author is partially supported by the NSF grant DMS-2055156

##  1\. The large ensemble problem

Given $S\subset[0,1]^{2}$, $F:S\to{\mathbb{C}}$ and $x\in{\mathbb{R}}^{3}$,
the restriction to the two dimensional paraboloid ${\mathbb{P}}^{2}$ is

$E_{S}F(x)=\int_{S}F(\xi)e((\xi,|\xi|^{2})\cdot x)d\xi.$

We write $EF$ for $E_{[0,1]^{2}}F$, and we denote by $B_{R}$ an arbitrary
ball/cube of radius/side length $R$ in ${\mathbb{R}}^{3}$. For context, we
recall the main result in [4].

Let ${\mathcal{Q}}_{R}$ denote the $\frac{1}{\sqrt{R}}$-squares $q$ in
$[0,1]^{2}$ of the form
$[\frac{j}{\sqrt{R}},\frac{j+1}{\sqrt{R}}]\times[\frac{i}{\sqrt{R}},\frac{i+1}{\sqrt{R}}]$.

######  Theorem 1.1.

For $2\leq p\leq 4$ we have

$\|EF\|_{L^{p}(B_{R})}\lesssim_{\epsilon}R^{\epsilon}(\sum_{q\in{\mathcal{Q}}_{R}}\|E_{q}F\|_{L^{p}(B_{R})}^{2})^{1/2}.$

To be fully precise, $L^{p}(B_{R})$ on the right hand side should be
$L^{p}(w_{B_{R}})$, where $w_{B_{R}}$ is a smooth approximation of
$1_{B_{R}}$. To keep notation simple, this will be ignored, as it has no
impact on our arguments. This inequality is known as (canonical) $l^{2}$
decoupling. It immediately implies the superficially weaker inequality called
(canonical) $l^{p}$ decoupling

$\|EF\|_{L^{p}(B_{R})}\lesssim_{\epsilon}R^{\frac{1}{2}-\frac{1}{p}+\epsilon}(\sum_{q\in{\mathcal{Q}}_{R}}\|E_{q}F\|_{L^{p}(B_{R})}^{p})^{1/p}.$

There are similar decouplings for other manifolds, but we only focus on
${\mathbb{P}}^{2}$ here. The phenomenon we are about to describe is already
complex in this case.

The scale $1/\sqrt{R}$ of the squares $q$ has a special relation to the scale
$R$ of the balls $B_{R}$. It allows the crucial use of parabolic rescaling in
the proof of Theorem 1.1. For this reason, the squares $q$ (or their lifts to
${\mathbb{P}}^{2}$) are sometimes called canonical caps. One notable variant
that has been considered is called small cap decoupling. This time $[0,1]^{2}$
is partitioned into smaller $\frac{1}{R^{\alpha}}$-squares,
$\frac{1}{2}<\alpha\leq 1$, called small caps. See [5] and [14].

Here, we go in the opposite direction and replace the squares $q$ with larger
sets. Considering just larger squares does not lead to anything new or
interesting. Instead, we will allow arbitrary unions of such squares. Examples
include thin rectangles and disconnected sets.

######  Definition 1.2.

A disjoint union of $\frac{1}{\sqrt{R}}$-squares $q\in{\mathcal{Q}}_{R}$ will
be called a large ensemble.

We kick-start our investigation with the following conjecture.

######  Conjecture 1.3 (Large ensemble decoupling for the paraboloid).

Let $N=O(R)$. Let $S_{1},\ldots,S_{N}$ be a partition of $[0,1]^{2}$ into
large ensembles. Then for $2\leq p\leq 3$ we have

$\|EF\|_{L^{p}(B_{R})}\lesssim_{\epsilon}R^{\epsilon}(\sum_{i=1}^{N}\|E_{S_{i}}F\|_{L^{p}(B_{R})}^{2})^{1/2}.$
(1)

The case $p=2$ follows from $L^{2}$ orthogonality. As we will see in the next
section, the case $p=3$ implies the full Restriction conjecture for
${\mathbb{P}}^{2}$. The same computations will show why the restriction $p\leq
3$ is needed. No interpolation is available in this general setting.

A well known consequence of Theorem 1.1 is the fact that for each $2\leq p\leq
4$ and each exponential sum ${\mathcal{E}}(x)=\sum_{q}a_{q}e(\xi_{q}\cdot x)$
associated with $1/\sqrt{R}$-separated points $\xi_{q}$ on ${\mathbb{P}}^{2}$
we have

$\|{\mathcal{E}}\|_{L^{p}(B_{R})}\approx\|a_{q}\|_{l^{2}}|B_{R}|^{1/p}.$

Here and in the following we use the notation $\approx$ to denote an equality
modulo $R^{\epsilon}$ multiplicative losses, for arbitrarily small
$\epsilon>0$. This immediately shows that when $EF$ is such an exponential
sum, (1) holds with $\lesssim$ replaced with $\approx$, in the larger range
$2\leq p\leq 4$, and for each $S_{i}$.

The reason we anticipate the validity of the conjecture for arbitrary $F$ is
because it is implied by the following reverse square function estimate,
widely believed to be true for $2\leq p\leq 3$

$\|EF\|_{L^{p}(B_{R})}\lesssim_{\epsilon}\|(\sum_{q\in{\mathcal{Q}}_{R}}|E_{q}F|^{2})^{1/2})\|_{L^{p}(B_{R})}.$
(2)

######  Theorem 1.4.

Assume (2) is true. Then Conjecture 1.3 holds.

###### Proof.

We first combine (2) with Minkowski’s inequality to get

$\|EF\|_{L^{p}(B_{R})}\lesssim(\sum_{i=1}^{N}\|(\sum_{q\in
S_{i}}|E_{q}F|^{2})^{1/2}\|^{2}_{L^{p}(B_{R})})^{1/2}.$

Note that

$\|(\sum_{q\in S_{i}}|E_{q}F|^{2})^{1/2}\|_{L^{p}(B_{R})}\approx\|(\sum_{q\in
S_{i}}|\eta_{R}E_{q}F|^{2})^{1/2}\|_{L^{p}({\mathbb{R}}^{3})},$

where $\eta_{R}$ is a smooth approximation of $1_{B_{R}}$, with compactly
supported Fourier transform. The Fourier transforms of the functions
$\eta_{R}E_{q}F$ are supported on a finitely overlapping family of
$1/\sqrt{R}$-cubes near ${\mathbb{P}}^{2}$.

To end, we use Rubio de Francia’s square function estimate for this collection
of cubes

$\|(\sum_{q\in
S_{i}}|\eta_{R}E_{q}F|^{2})^{1/2}\|_{L^{p}({\mathbb{R}}^{3})}\lesssim\|\sum_{q\in
S_{i}}\eta_{R}E_{q}F\|_{L^{p}({\mathbb{R}}^{3})}\approx\|E_{S_{i}}F\|_{L^{p}(B_{R})}.$

∎

In Section 2 we prove that Conjecture 1.3 implies the Restriction conjecture
for ${\mathbb{P}}^{2}$. Since proving it for arbitrary partitions is out of
reach, in Section 3 we introduce methods aimed to solve a particular case of
interest.

######  Acknowledgment.

I am grateful to Larry Guth for providing me with the probabilistic heuristics
for Corollary 2.2. I thank Nets Katz for suggesting the name for large
ensemble decoupling. The author has enjoyed conversations on this topic with
Hong Wang and Aric Wheeler.

##  2\. Conjecture 1.3 implies the Restriction Conjecture

Here is the main result in this section. A union of $M$ canonical squares $q$
is called an $M$-set.

######  Theorem 2.1.

Fix $0<\gamma\leq 1$. Assume that the following large ensemble $l^{p}$
decoupling holds for some $p\in[2,4]$, $M\sim R^{1-\gamma}$ (and all
$\epsilon>0$, all $R\gtrsim 1$)

$\|EF\|_{L^{p}(B_{R})}^{p}\lesssim_{\epsilon}R^{\epsilon}(\frac{R}{M})^{\frac{p}{2}-1}\sum_{j=1}^{R/M}\|E_{P_{j}}F\|_{L^{p}(B_{R})}^{p},$
(3)

for all $F$ and all partitions ${\mathcal{P}}$ of ${\mathcal{Q}}_{R}$ into
$M$-sets $P_{j}$. Then for each $\|F\|_{\infty}\lesssim R$ we have

$\int_{B_{\sqrt{R}}}|EF|^{p}\lesssim_{\epsilon}R^{\epsilon}(R^{\frac{3+p}{2}}+R^{3}(\frac{R}{M})^{\frac{p}{2}-1}).$

Before we prove the theorem, we reveal its significance with two immediate
consequences.

######  Corollary 2.2.

Assume (3) holds true for $p=3$ and all $\gamma$ arbitrarily close to $0$.
Then the Restriction conjecture for the paraboloid holds: for each $F$ we have

$\int_{B_{\sqrt{R}}}|EF|^{3}\lesssim_{\epsilon}R^{\epsilon}\|F\|_{\infty}^{3}.$

######  Corollary 2.3.

Assume (3) holds true for some $p\in(3,4]$ and $M\sim R^{\frac{4-p}{p-2}}$.
Then the Restriction conjecture for the paraboloid holds in $L^{p}$: for each
$F$ we have

$\int_{B_{\sqrt{R}}}|EF|^{p}\lesssim_{\epsilon}R^{\epsilon}\|F\|_{\infty}^{p}.$

Moreover, if $\Delta>0$ then (3) is false for $p\in(3,4]$, $M\sim
R^{\frac{4-p}{p-2}+\Delta}$ and some partition into $M$-sets.

To see the last part of Corollary 2.3, note that if (3) held for some $M\sim
R^{\frac{4-p}{p-2}+\Delta}$, then Theorem 2.1 would imply the false inequality
(use $F=1_{[0,1]^{2}}$)

$\int_{B_{\sqrt{R}}}|EF|^{p}\lesssim R^{-\beta}\|F\|_{\infty}^{p},$

for some $\beta>0$.

The next two lemmas establish average $l^{2}$ and $l^{4}$ estimates, via
elementary expansion and counting arguments. These will later be interpolated
to get $l^{3}$ estimates. Such pointwise estimates are then integrated on
$B_{R}$ to get $L^{3}$ estimates. The idea is that averaging over partitions
produces square root cancellation. When combined with the conjectured
decoupling into large ensembles, this leads to restriction estimates as
described above.

In the following, sums are taken over all partitions ${\mathcal{P}}$ of
$\\{1,\ldots,R\\}$ into sets $P_{j}$ with uniform size $M\leq R$. Partitions
are always considered unordered.

######  Lemma 2.4.

For each $a_{i}\in{\mathbb{C}}$, $1\leq i\leq R$

$\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{2}\lesssim\frac{M}{R}|\sum_{i}a_{i}|^{2}+\sum_{i}|a_{i}|^{2}.$

###### Proof.

We have

$\sum_{{\mathcal{P}}}\sum_{P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{2}=\sum_{{\mathcal{P}}}\sum_{i}|a_{i}|^{2}+\sum_{i_{1}\not=i_{2}}\\#{\mathcal{P}}(i_{1},i_{2})\overline{a_{i_{1}}}a_{i_{2}}$

where ${\mathcal{P}}(i_{1},i_{2})$ are the special partitions that put
${i_{1}},{i_{2}}$ into the same cell. Note that we have

$\frac{1}{(R/M)!}{R\choose M}{R-M\choose M}\ldots{2M\choose M}$

partitions ${\mathcal{P}}$. Indeed, we first select the first cell, which can
be done in $R\choose M$ ways. The second to last cell can be selected in
$2M\choose M$ ways. We need to divide with $(R/M)!$, as the partition is not
ordered.

The number of partitions ${\mathcal{P}}(i_{1},i_{2})$ is

$\frac{1}{(\frac{R}{M}-1)!}{R-2\choose M}{R-M-2\choose M}\ldots{2M-2\choose
M}.$

We select the first $\frac{R}{M}-1$ cells from
$[1,R]\setminus\\{i_{1},i_{2}\\}$, and put $i_{1},i_{2}$ in the small cell
left in the end. The number of selected cells needs to be divided by the
number of permutations. Note that

$\\#{\mathcal{P}}(i_{1},i_{2})=\\#{\mathcal{P}}\times\frac{R}{M}\frac{M(M-1)}{R(R-1)}\sim\\#{\mathcal{P}}M/R.$

We conclude that

$\sum_{{\mathcal{P}}}\sum_{P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{2}=$
$\\#{\mathcal{P}}(\sum_{i}|a_{i}|^{2}+\frac{M-1}{R-1}\sum_{i_{1}\not=i_{2}}\overline{a_{i_{1}}}a_{i_{2}})=\\#{\mathcal{P}}(\frac{R-M}{R-1}\sum_{i}|a_{i}|^{2}+\frac{M-1}{R-1}|\sum_{i}a_{i}|^{2}).$

∎

######  Lemma 2.5.

For each $a_{i}\in{\mathbb{C}}$

$\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{\;P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{4}\lesssim$
$\frac{M^{3}}{R^{3}}|\sum_{i}a_{i}|^{4}+\frac{M^{2}}{R^{2}}\sum_{i}|a_{i}|^{2}|\sum_{i}a_{i}|^{2}+\frac{M}{R}\sum_{i}|a_{i}|^{3}\sum_{i}|a_{i}|+\sum_{i}|a_{i}|^{4}.$

###### Proof.

The expansion of the term on the left will contain four types of quadruple
products.

First, there are the terms
$a_{i_{1}}\overline{a_{i_{2}}}a_{i_{3}}\overline{a_{i_{4}}}$ with
$i_{1}\not=i_{2}\not=i_{3}\not=i_{4}$. Reasoning as in the previous lemma,
each such term appears roughly $M^{3}\\#{\mathcal{P}}/R^{3}$ many times in the
summation, as this is the number of partitions that place the four indices in
the same cell.

Second, there are the terms of the form
$a_{i_{1}}\overline{a_{i_{1}}}a_{i_{2}}\overline{a_{i_{3}}}$ (and their
permutations) with $i_{1}\not=i_{2}\not=i_{3}$. These collect all products
where exactly three of the four indices are distinct. Each such term appears
roughly $M^{2}\\#{\mathcal{P}}/R^{2}$ many times.

Third, we will have products containing two distinct indices, of the form
$a_{i_{1}}\overline{a_{i_{1}}}a_{i_{2}}\overline{a_{i_{2}}}$,
$a_{i_{1}}\overline{a_{i_{1}}}a_{i_{1}}\overline{a_{i_{2}}}$ and their
permutations. Each of those appear roughly $M\\#{\mathcal{P}}/R$ many times.

Finally, each term $|a_{i}|^{4}$ appears exactly $\\#{\mathcal{P}}$ many
times.

The contributions from the first category is dominated by (we add the last
term to “complete the square”)

$\lesssim\frac{M^{3}\\#{\mathcal{P}}}{R^{3}}(|\sum_{i}a_{i}|^{4}+|\sum_{i_{1},i_{2},i_{3},i_{4}\atop{\text{not
all four
distinct}}}a_{i_{1}}\overline{a_{i_{2}}}a_{i_{3}}\overline{a_{i_{4}}}|).$

We also have the following estimate for the term we added

$\frac{M^{3}\\#{\mathcal{P}}}{R^{3}}|\sum_{i_{1},i_{2},i_{3},i_{4}\atop{\text{not
all four
distinct}}}a_{i_{1}}\overline{a_{i_{2}}}a_{i_{3}}\overline{a_{i_{4}}}|\leq$
$\frac{M^{3}\\#{\mathcal{P}}}{R^{3}}(|\sum_{i_{1},i_{2},i_{3},i_{4}\atop{\text{three
distinct
indices}}}a_{i_{1}}\overline{a_{i_{2}}}a_{i_{3}}\overline{a_{i_{4}}}|+|\sum_{i_{1},i_{2},i_{3},i_{4}\atop{\text{two
distinct
indices}}}a_{i_{1}}\overline{a_{i_{2}}}a_{i_{3}}\overline{a_{i_{4}}}|+\sum_{i}|a_{i}|^{4}).$

These contributions are absorbed by the remaining three terms, as the exponent
three in the coefficient $(M/R)^{3}$ here is larger than those in the
remaining cases.

Indeed, the contribution from the second case is

$\lesssim\frac{M^{2}\\#{\mathcal{P}}}{R^{2}}|\sum_{i_{1},i_{2},i_{3},i_{4}\atop{\text{three
distinct
indices}}}a_{i_{1}}\overline{a_{i_{2}}}a_{i_{3}}\overline{a_{i_{4}}}|.$

Let us analyze the case $i_{1}=i_{2}$, the other ones are similar. Then,
completing the square again

$\frac{M^{2}\\#{\mathcal{P}}}{R^{2}}|\sum_{i_{1},i_{3},i_{4}\atop{{i_{1}\not=i_{3}\not=i_{4}}}}|a_{i_{1}}|^{2}a_{i_{3}}\overline{a_{i_{4}}}|\leq\frac{M^{2}\\#{\mathcal{P}}}{R^{2}}(\sum_{i}|a_{i}|^{2}|\sum_{i}a_{i}|^{2}+\sum_{i_{1},i_{2},i_{3}\atop{\text{not
all three distinct}}}|a_{i_{1}}|^{2}a_{i_{3}}\overline{a_{i_{4}}}).$

As before, the term we added is absorbed by the contributions of the remaining
two cases.

Indeed, the contribution from the third case is

$\frac{M\\#{\mathcal{P}}}{R}|\sum_{i_{1},i_{2},i_{3},i_{4}\atop{\text{only two
distinct
indices}}}a_{i_{1}}\overline{a_{i_{2}}}a_{i_{3}}\overline{a_{i_{4}}}|.$

This is dominated by

$\frac{M\\#{\mathcal{P}}}{R}((\sum_{i}|a_{i}|^{2})^{2}+\sum_{i_{1}}|a_{i_{1}}|^{3}\sum_{i_{2}}|a_{i_{2}}|).$

The second term dominates because of Cauchy–Schwarz. The contribution from the
fourth case is

$\\#{\mathcal{P}}\sum_{i}|a_{i}|^{4}.$

∎

Next, we interpolate.

######  Corollary 2.6.

Assume $|a_{i}|\lesssim 1$ and let $L=\sum_{i=1}^{R}|a_{i}|$,
$S=|\sum_{i=1}^{R}a_{i}|$. Then for $2\leq p\leq 4$

$\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{\;P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{p}\lesssim(\frac{M}{R})^{\frac{p}{2}-1}L^{p/2}+L+(\frac{M}{R})^{p/2}S^{p}.$

###### Proof.

Recall that

$\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{2}\lesssim\frac{M}{R}S^{2}+L$

and

$\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{\;P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{4}\lesssim\frac{M^{3}}{R^{3}}S^{4}+\frac{M^{2}}{R^{2}}LS^{2}+\frac{M}{R}L^{2}+L.$

We combine these with the inequality
$\|\cdot\|_{l^{p}}^{p}\leq\|\cdot\|_{l^{2}}^{4-p}\|\cdot\|_{l^{4}}^{2p-4}$.

Case 1. If $L\geq\frac{M}{R}S^{2}$ then we get

$\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{\;P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{p}\lesssim
L^{2-\frac{p}{2}}(\frac{M}{R}L^{2}+L)^{\frac{p}{2}-1}\lesssim(\frac{M}{R})^{\frac{p}{2}-1}L^{p/2}+L.$

Case 2. If $L\leq\frac{M}{R}S^{2}$ then the term $\|\cdot\|_{l^{4}}^{4}$ is
dominated by $\frac{M^{3}}{R^{3}}S^{4}+L$. Since $S\leq L$ and $M\leq R$, we
must have that $L\geq 1$. Thus $L\leq L^{2}\leq\frac{M^{2}}{R^{2}}S^{4}$, and
we may further dominate $\|\cdot\|_{l^{4}}^{4}$ by $\frac{M^{2}}{R^{2}}S^{4}$.
We get

$\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{\;P_{j}\in{\mathcal{P}}}|\sum_{i\in
P_{j}}a_{i}|^{p}\lesssim(\frac{M}{R}S^{2})^{2-\frac{p}{2}}(\frac{M^{2}}{R^{2}}S^{4})^{\frac{p}{2}-1}\lesssim(\frac{M}{R})^{p/2}S^{p}.$

∎

We now verify Theorem 2.1.

###### Proof.

The first part of the argument follows the lines of [9]. We may assume
$B_{\sqrt{R}}$ is centered at the origin. Due to the smoothing effect of the
convolution with the Fourier transform of the characteristic function of this
ball, we may assume $F$ is constant on each $q$. For such $F$, we will in fact
prove the superficially stronger inequality on the larger ball $B_{R}$
centered at the origin

$\int_{B_{R}}|EF|^{p}\lesssim_{\epsilon}R^{\epsilon}(R^{\frac{3+p}{2}}+R^{3}(\frac{R}{M})^{\frac{p}{2}-1}).$

On $B_{R}$, each $E_{q_{i}}F$ is a single wave packet $\Phi_{i}$ that is
spatially localized near a $(R,R^{1/2},R^{1/2})$-tube $T_{i}$ through the
origin, and has $L^{\infty}$ norm $\lesssim 1$. We may write this as

$E_{q_{i}}F(x)=\Phi_{i}(x)1_{R^{\delta}T_{i}}(x)+O_{\delta}(R^{-1000}),\;\;x\in
B_{R},$

for $\delta$ as small as we wish. The first key inequality we use is the well-
known bush estimate

$\|\sum_{i=1}^{R}1_{R^{\delta}T_{i}}\|_{r}\lesssim
R^{1+\frac{3}{2r}+O(\delta)}$

valid for $r\geq 3/2$. This follows from an easy computation. It shows that
the main contribution to the integral comes from the core $B_{\sqrt{R}}$ where
all tubes intersect. This justifies the fact that moving integration from this
smaller ball to $B_{R}$ is not lossy.

We average (3) over all partitions ${\mathcal{P}}$ into $M$-sets to get

$\|EF\|_{L^{p}(B_{R})}^{p}\leq
C_{R}(\frac{R}{M})^{\frac{p}{2}-1}\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{P_{j}\in{\mathcal{P}}}^{R/M}\|\sum_{q_{i}\in
P_{j}}E_{q_{i}}F\|_{L^{p}(B_{R})}^{p},$ (4)

with $C_{R}\lesssim_{\epsilon}R^{\epsilon}$. Writing
$a_{i}(x)=\Phi_{i}(x)1_{R^{\delta}T_{i}}(x)$, we have

$(\frac{R}{M})^{\frac{p}{2}-1}\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{P_{j}\in{\mathcal{P}}}^{R/M}|\sum_{q_{i}\in
P_{j}}E_{q_{i}}F(x)|^{p}=(\frac{R}{M})^{\frac{p}{2}-1}\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{P_{j}\in{\mathcal{P}}}^{R/M}|\sum_{q_{i}\in
P_{j}}a_{i}(x)|^{p}+O_{\delta}(R^{-100}).$

Write

$S(x)=|\sum_{i}a_{i}(x)|=|EF(x)|+O_{\delta}(R^{-100})$ (5)

and

$L(x)=\\#\\{i:\;x\in 1_{R^{\delta}T_{i}}\\}.$

Corollary 2.6 implies that

$(\frac{R}{M})^{\frac{p}{2}-1}\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}}\sum_{P_{j}\in{\mathcal{P}}}^{R/M}|\sum_{q_{i}\in
P_{j}}a_{i}(x)|^{p}\lesssim
L(x)^{p/2}+(\frac{R}{M})^{\frac{p}{2}-1}L(x)+\frac{M}{R}S(x)^{3}.$

We integrate this over $B_{R}$ and combine with (4) and (5) to conclude that

$\|EF\|_{L^{p}(B_{R})}^{p}\lesssim
C_{R}(\int_{B_{R}}L^{p/2}+(\frac{R}{M})^{\frac{p}{2}-1}\int
L+\frac{M}{R}\|EF\|_{L^{3}(B_{R})}^{3})+O_{\delta}(R^{-10}).$

If $M\ll R/C_{R}$ this leads to

$\|EF\|_{L^{p}(B_{R})}^{p}\lesssim
C_{R}(\int_{B_{R}}L^{p/2}+(\frac{R}{M})^{\frac{p}{2}-1}\int_{B_{R}}L)+O_{\delta}(R^{-10}).$

Using the bush estimate and the trivial $\int_{B_{R}}L\lesssim
R^{3+O(\delta)}$, we get

$\|EF\|_{L^{p}(B_{R})}^{p}\lesssim_{\epsilon}R^{\epsilon}(R^{\frac{3+p}{2}}+R^{3}(\frac{R}{M})^{\frac{p}{2}-1}),$

by choosing $\delta$ small. ∎

The conclusion of Corollary 2.2 remains true if we average over a more
restricted collection of partitions of the following form. We fix $M$
collections $S_{k}=S_{k}(M)$ each consisting of $R/M$ squares $q$, in such a
way that the $S_{k}$ form a partition of all squares. The partitions
${\mathcal{P}}$ we will use in the next theorem consist of $M$-sets $P_{j}$,
each having exactly one element from each $S_{k}$. There are
$(\frac{R}{M}!)^{M-1}$ such partitions. The proof is left as an exercise to
the reader.

######  Theorem 2.7.

Assume that for all $M$ sufficiently close to $R$ we have

$\|EF\|_{L^{3}(B_{R})}^{3}\lesssim_{\epsilon}R^{\epsilon}\frac{1}{\\#{\mathcal{P}}}\sum_{{\mathcal{P}}=\\{P_{j}\\}}(\frac{R}{M})^{1/2}\sum_{j=1}^{R/M}\|E_{P_{j}}F\|_{L^{3}(B_{R})}^{3}.$

Then the Restriction conjecture for ${\mathbb{P}}^{2}$ holds.

The situation is even more dramatic for $l^{2}$ decoupling.

######  Theorem 2.8.

Fix $\beta\in(0,1)$. Assume (1) holds for all partitions into $R^{\beta}$-sets
$S_{i}$ when $p=3$. Then the Restriction conjecture for ${\mathbb{P}}^{2}$
holds.

###### Proof.

Fix $\delta$ such that $\frac{\beta-1}{2}+\delta\leq 0$. Write
$R^{\prime}=R^{\beta+2\delta}$ and
$M=R^{\beta}=(R^{\prime})^{\frac{\beta}{\beta+2\delta}}$. We consider $F$
supported on $[0,R^{\frac{\beta-1}{2}+\delta}]^{2}$. Inequality (1) and
Hölder’s inequality implies that for each such $F$ and each partition
${\mathcal{P}}=\\{P_{j}\\}$ of the collection of $1/\sqrt{R}$-squares $q$ in
$[0,R^{\frac{\beta-1}{2}+\delta}]^{2}$ into $M$-sets $P_{j}$ we have

$\|EF\|_{L^{3}(B_{R})}^{3}\lesssim_{\epsilon}R^{\epsilon}(\frac{R^{\prime}}{M})^{1/2}\sum_{j=1}^{R^{\prime}/M}\|E_{P_{j}}F\|_{L^{3}(B_{R})}^{3}.$

Using parabolic rescaling
$\xi=(\xi_{1},\xi_{2})\to\xi^{\prime}=R^{\frac{1-\beta}{2}-\delta}(\xi_{1},\xi_{2})$
on the ball $B_{R}=[-R,R]^{3}$, this can be written as

$\|E\tilde{F}\|_{L^{3}([-R^{\frac{\beta+1}{2}+\delta},R^{\frac{\beta+1}{2}+\delta}]^{2}\times[-R^{\prime},R^{\prime}])}^{3}\lesssim_{\epsilon}(R^{\prime})^{\epsilon}(\frac{R^{\prime}}{M})^{1/2}\sum_{j=1}^{R^{\prime}/M}\|E_{P_{j}}\tilde{F}\|_{L^{3}([-R^{\frac{\beta+1}{2}+\delta},R^{\frac{\beta+1}{2}+\delta}]^{2}\times[-R^{\prime},R^{\prime}])}^{3}.$

This holds true for all $\tilde{F}:[0,1]^{2}\to{\mathbb{C}}$ and all
partitions of $[0,1]^{2}$ into $M$-sets $P_{j}$ consisting of
$1/\sqrt{R^{\prime}}$-squares $q^{\prime}$. We will use the superficially
weaker consequence

$\|E\tilde{F}\|_{L^{3}([-R^{\prime},R^{\prime}]^{3})}^{3}\lesssim_{\epsilon}(R^{\prime})^{\epsilon}(\frac{R^{\prime}}{M})^{1/2}\sum_{j=1}^{R^{\prime}/M}\|E_{P_{j}}\tilde{F}\|_{L^{3}({\mathbb{R}}^{2}\times[-R^{\prime},R^{\prime}])}^{3}.$
(6)

The rest of the argument is identical to the one for Theorem 2.1. We consider
an arbitrary $\tilde{F}$ with $\|\tilde{F}\|_{\infty}\lesssim R^{\prime}$ and
prove that

$\int_{[0,\sqrt{R^{\prime}}]^{3}}|\tilde{F}|^{3}\lesssim_{\epsilon}(R^{\prime})^{3+\epsilon}.$

We may assume $\tilde{F}$ is constant on each $q^{\prime}$. For such
$\tilde{F}$ we prove the superficially stronger inequality

$\|E\tilde{F}\|_{L^{3}([-R^{\prime},R^{\prime}]^{3})}^{3}\lesssim_{\epsilon}(R^{\prime})^{3+\epsilon}.$
(7)

We then average (6) over all partitions, and proceed as before to find

$\int_{[-R^{\prime},R^{\prime}]^{3}}|E\tilde{F}|^{p}\lesssim_{\epsilon}(R^{\prime})^{\epsilon}((R^{\prime})^{3}+(R^{\prime})^{3}(\frac{R^{\prime}}{M})^{3/2}).$

We have used that

$\|E_{P_{j}}\tilde{F}\|_{L^{3}({\mathbb{R}}^{2}\times[-R^{\prime},R^{\prime}])}^{3}\approx\|E_{P_{j}}\tilde{F}\|_{L^{3}([-R^{\prime},R^{\prime}]^{3})}^{3}.$

This is because the mass of $E_{P_{j}}\tilde{F}$ is concentrated in a tube
that lives in $[-R^{\prime},R^{\prime}]^{3}$.

Recall that $R^{\prime}/M=R^{2\delta}$. The estimate above holds uniformly for
each $R^{\prime}\gg 1$ and $\delta$. Inequality (7) is seen to hold by letting
$\delta,\epsilon\to 0$.

∎

When $M$ is small, the validity of (3) seems much weaker than the full
Restriction conjecture. For example, the case $M=1$ is Theorem 1.1. The value
$M=R^{1/2}$ is particularly appealing, as (3) for this $M$ and $p=10/3$
implies (via Corollary 2.3) the sharp restriction estimate for $p=10/3$. This
exponent, initially established by Tao [19], was a landmark threshold until it
was reproved and slightly improved in [6]. This suggests that perhaps (3)
could be proved with existing technology when $M$ is $R^{1/2}$ (or smaller).
We caution that the $l^{2}$ version of this when $p=3$ implies the Restriction
conjecture, see Theorem 2.8.

######  Conjecture 2.9.

Assume each $S_{i}$ is a $\sqrt{R}$-set. The inequality

$\|EF\|_{L^{p}(B_{R})}\lesssim_{\epsilon}R^{\frac{1}{2}(\frac{1}{2}-\frac{1}{p})+\epsilon}(\sum_{i=1}^{R^{1/2}}\|E_{S_{i}}F\|_{L^{p}(B_{R})}^{p})^{1/p}$
(8)

holds for $2\leq p\leq 10/3$.

We offer evidence for the conjecture by verifying the case when
$EF=\sum_{T}EF_{T}$, with $EF_{T}$ a single wave packet corresponding to each
$q$ (so $T=T_{q}$), $|EF_{T}|\approx 1_{T}$ on $B_{R}$. We start with the
endpoint $p=10/3$.

First, an easy consequence of Tao’s bilinear theorem [19] is the reverse
square function estimate (see e.g. Exercise 7.25 in [10])

$\|EF\|_{L^{10/3}(B_{R})}\lesssim_{\epsilon}R^{1/20+\epsilon}\|(\sum_{T}|EF_{T}|^{2})^{1/2}\|_{L^{10/3}(B_{R})}.$

This inequality holds true for arbitrary $F$, but is sharp in our case, when
all tubes intersect.

Next, note that

$\|(\sum_{T}|EF_{T}|^{2})^{1/2}\|_{L^{10/3}(B_{R})}\approx\|\sum_{T}1_{T}\|_{L^{5/3}}^{1/2}.$

Then, using Wolff’s Kakeya estimate [20] (there is one tube per direction) we
have

$\|\sum_{T}1_{T}\|_{5/3}\lesssim_{\epsilon}\|\sum_{T}1_{T}\|_{L^{1}}^{3/5}R^{1/10+\epsilon}.$

Putting these together we get

$\|EF\|_{L^{10/3}(B_{R})}\lesssim_{\epsilon}\|\sum_{T}1_{T}\|_{L^{1}}^{3/10}R^{1/10+\epsilon}.$

Finally, note that by using Rubio de Francia’s square function estimate as
before we find

$\|E_{S_{i}}F\|_{L^{10/3}(B_{R})}\gtrsim\|(\sum_{T=T_{q}\atop{q\subset
S_{i}}}|EF_{T}|^{2})^{1/2}\|_{L^{10/3}(B_{R})}\approx\|\sum_{T=T_{q}\atop{q\subset
S_{i}}}1_{T}\|_{L^{5/3}}^{1/2}\gtrsim\|\sum_{T=T_{q}\atop{q\subset
S_{i}}}1_{T}\|_{L^{1}}^{3/10}.$

The last inequality is due to the fact that the sum is either zero or greater
than 1.

When $p<10/3$, the inequality loses strength. The upper bound
$\|EF\|_{L^{p}(B_{R})}\lesssim R^{\frac{5}{2p}+\frac{1}{4}}$ follows by
interpolating the $L^{10/3}$ bound $O(R)$ from above with the trivial $L^{2}$
bound $O(R^{3/2})$. This estimate is not sharp, except for the endpoints, but
suffices for our purpose. More precisely, the Restriction conjecture
anticipates that $\|EF\|_{L^{p}(B_{R})}\lesssim_{\epsilon}R^{1+\epsilon}$ for
$p\geq 3$.

Lower bounds for $\|E_{S_{i}}F\|_{L^{p}(B_{R})}$ follow as before, and produce
the lower bound $R^{\frac{5}{2p}+\frac{1}{4}}$ for the right hand side of (8).

In the next section, we prove an unconditional result for $\sqrt{R}$-sets that
are one dimensional in nature.

##  3\. An unconditional large ensemble decoupling

Here is the main result in this section.

######  Theorem 3.1.

Let $\sqrt{R}\lesssim N\leq R$. Let $S_{1},\ldots,S_{N}$ be a partition of
$[0,1]^{2}$ into large ensembles, such that each square $Q\subset[0,1]^{2}$
with side length $L\geq R^{-1/2}$ intersects at most $O(LN)$ of these sets.
Then for each $F:[0,1]^{2}\to{\mathbb{C}}$ and $2\leq p\leq 3$ we have

$\|EF\|_{L^{p}(B_{R})}\lesssim_{\epsilon}R^{\epsilon}N^{\frac{1}{2}-\frac{1}{p}}(\sum_{i=1}^{N}\|E_{S_{i}}F\|_{L^{p}(B_{R})}^{p})^{1/p}.$

The proof relies on recoupling. We formulate it in a way that fits our needs
here.

######  Theorem 3.2 (Recoupling).

Assume $F$ is supported on pairwise disjoint squares
$\alpha_{1},\ldots,\alpha_{M}$ with side length $1/K$. Then for each ball $B$
with radius at least $K$ and for each $p\geq 2$ we have

$(\sum_{i=1}^{M}\|E_{\alpha_{i}}F\|_{L^{p}(B)}^{p})^{1/p}\lesssim\|EF\|_{L^{p}(B)}.$

###### Proof.

The result is immediate when $p\in\\{2,\infty\\}$. Use
$l^{2}(L^{2})-l^{\infty}(L^{\infty})$ interpolation. Alternatively, use Rubio
de Francia’s square function estimate, as in the proof of Theorem 1.4.

∎

We also use the following unconditional version of Theorem 1.4. Its proof
mirrors the one of Theorem 1.4, using the trilinear reverse square function
estimate in [2].

######  Theorem 3.3.

The trilinear version of Conjecture 1.3 holds. More precisely, if
$F=F_{1}+F_{2}+F_{3}$ and $F_{i}$ are supported transversely, then for $2\leq
p\leq 3$

$\|(\prod_{i}EF_{i})^{1/3}\|_{L^{p}(B_{R})}\lesssim_{\epsilon}R^{\epsilon}(\sum_{i=1}^{N}\|E_{S_{i}}F\|_{L^{p}(B_{R})}^{2})^{1/2}.$

###### Proof.

(of Theorem 3.1)

Let $C_{R}$ be the best constant such that

$\|EF\|_{L^{p}(B_{R})}\leq
C_{R}N^{\frac{1}{2}-\frac{1}{p}}(\sum_{i=1}^{N}\|E_{S_{i}}F\|_{L^{p}(B_{R})}^{p})^{1/p}$

holds for all $F$, $B_{R}$, $N$ and $S_{i}$ as in the statement of Theorem
3.1. Fix $K=O(1)$, large enough.

We follow a standard Bourgain–Guth argument, see [6]. Split $[0,1]^{2}$ into
$1/K$-squares $\alpha$. Consider a finitely overlapping cover of $B_{R}$ with
balls $B_{K}$. For each $B_{K}\subset B_{R}$, there is a flat $1\times
1/K$-strip $S$ (that is, the collection of all $1/K$-squares $\alpha$ whose
slight enlargements intersect a given line) inside $[0,1]^{2}$, and there are
transverse (that is, any three points from the three squares determine an area
$\gtrsim_{K}1$) $1/K$-squares $\alpha_{1},\alpha_{2},\alpha_{3}$ such that

$|EF(x)|\lesssim
K^{O(1)}|\prod_{i=1}^{3}E_{\alpha_{i}}F(x)|^{1/3}+|\sum_{\alpha\subset
S}E_{\alpha}F(x)|,\;x\in B_{K}.$

Small cap $l^{p}(L^{p})$ decoupling for the parabola ([11]) gives

$\int_{B_{K}}|\sum_{\alpha\subset S}E_{\alpha}F|^{p}\lesssim
K^{\frac{p}{2}-1}\sum_{\alpha\subset S}\int_{B_{K}}|E_{\alpha}F|^{p}.$

Summing up over $B_{K}$ we get

$\|EF\|_{L^{p}(B_{R})}^{p}\lesssim_{\epsilon}K^{O(1)}R^{\epsilon}(\sum_{i=1}^{N}\|E_{S_{i}}F\|_{L^{p}(B_{R})}^{2})^{p/2}+K^{\frac{p}{2}-1}\sum_{\alpha\subset[0,1]^{2}}\int_{B_{R}}|E_{\alpha}F|^{p}.$
(9)

The first term is the transverse contribution, estimated using Theorem 3.3 and
the triangle inequality to rebuild the fractured pieces $S_{i}$ (no need for
recoupling, as further losses in $K$ are harmless for this term).

For the second term, parabolic rescaling gives

$\int_{B_{R}}|E_{\alpha}F|^{p}\lesssim
C_{R/K^{2}}^{p}(\frac{N}{K})^{\frac{p}{2}-1}\sum_{S_{i}\cap\alpha\not=\emptyset}\int_{B_{R}}|E_{S_{i}\cap\alpha}F|^{p}.$

Here $S_{i}\cap\alpha$ is the union of the $1/\sqrt{R}$-squares lying inside
$\alpha$. Under rescaling $\alpha$ becomes $[0,1]^{2}$, and the sets $S_{i}$
that intersect $\alpha$ become unions of $K/\sqrt{R}$-squares. There are
$O(N/K)$ such sets, and they satisfy the induction hypothesis relative to
squares $Q$. The ball $B_{R}$ becomes a tubular region, covered by a finitely
overlapping union of balls of radius $R/K^{2}$.

Thus, the contribution from the second term in (9) is

$\lesssim
C_{R/K^{2}}^{p}N^{\frac{p}{2}-1}\sum_{S_{i}}\int_{B_{R}}\sum_{\alpha}|E_{\alpha\cap
S_{i}}F|^{p},$

which is dominated via recoupling by

$C_{R/K^{2}}^{p}N^{\frac{p}{2}-1}\sum_{S_{i}}\int_{B_{R}}|E_{S_{i}}F|^{p}.$

We conclude that

$C_{R}\leq C_{\epsilon}K^{O(1)}R^{\epsilon}+C_{1}C_{R/K^{2}},$

with $C_{1}$ independent of $K,R$. Iterating this $R\to R/K^{2}\to
R/K^{4}\ldots$ leads to $C_{R}\lesssim_{\epsilon}1$.

∎

Recoupling is typically lossy, more so than decoupling. The reason it is
harmless in this argument is because its loss is compensated by the gain in
the Bourgain–Guth argument, that comes from the one dimensional reduction to
strips.

One example where Theorem 3.1 is applicable with $N\sim\sqrt{R}$ is when each
$S_{i}$ consists of all the squares $q$ intersecting some curve $\gamma_{i}$.
We assume each $Q$ with side length $L$ intersects at most $L\sqrt{R}$ such
curves. If these curves are consecutive parallel lines at distance $\sim
1/\sqrt{R}$ from each other, the large ensembles $S_{i}$ are flat strips. But
the curves can also be e.g. concentric circles.

We close this section with a few interesting questions that the earlier
argument seems insufficient to address.

######  Question 3.4.

Prove $l^{2}(L^{3})$ decoupling into flat strips.

Conjecture 2.9 may be more manageable for flat strips.

######  Question 3.5.

Prove $l^{p}(L^{p})$ decoupling into flat strips for $3<p\leq 10/3.$

Let us see why $p$ cannot be larger than $10/3$. Note that this does not
follow from Corollary 2.3. Indeed, the optimal range of $p$ for a given
partition into $\sqrt{R}$-sets is sensitive to the geometry of the partition.
For example, (3) holds in the range $2\leq p\leq 4$ when each $P_{j}$ is an
$R^{-1/4}$-square. We test the inequality for flat strips with
$F=R1_{[0,1]^{2}}$. Each $E_{q}F$ is a single wave packet with magnitude $\sim
1$, so $|E_{q}F|\approx 1_{T_{q}}$, for some tube $T_{q}$ with radius
$R^{1/2}$ and length $R$, containing the origin. By constructive interference
we have $|EF(x)|\sim R$ for $|x|\lesssim 1$, so

$\|EF\|_{L^{p}(B_{R})}\gtrsim R.$

Using Cordoba’s inequality for the parabola above the strip $S_{i}$, followed
by the bush estimate for tubes, we find that for $p\leq 4$

$\|E_{S_{i}}F\|_{L^{p}(B_{R})}\lesssim_{\epsilon}R^{\epsilon}\|(\sum_{q\subset
S_{i}}|E_{q}F|^{2})^{1/2}\|_{L^{p}(B_{R})}\sim R^{\epsilon}\|\sum_{q\in
S_{i}}1_{T_{q}}\|_{p/2}^{1/2}\sim R^{\epsilon}\|\sum_{q\in
S_{i}}1_{T_{q}}\|_{1}^{1/p}\sim R^{\frac{5}{2p}+\epsilon}.$

Apart from $R^{\epsilon}$, this computation is sharp. Then (3) forces that

$R^{p}\lesssim_{\epsilon}R^{\epsilon}(\sqrt{R})^{p/2}R^{5/2},$

or $p\leq 10/3$.

The sets covered by Theorem 3.1 are “one-dimensional” in nature. The following
spread-out case seems to lie at the opposite end of the spectrum.

Split $[0,1]^{2}$ into $R^{1/2}$ squares $Q$ with side length $R^{-1/4}$. Each
$Q$ is the union of $R^{1/2}$ canonical squares $q$. We choose each $S_{i}$ to
contain exactly one $q$ from each $Q$. This is connected to Theorem 2.7.

######  Question 3.6.

Prove $l^{3}(L^{3})$ decoupling for these large ensembles.

The following is worth noticing, as it shows that the contribution from each
large ensemble is “computable”. As a consequence of Theorem 1.1 and Rubio de
Francia’s square function inequality, we have for $2\leq p\leq 4$ and each
ball $B_{\sqrt{R}}$

$\|E_{S_{i}}F\|_{L^{p}(B_{\sqrt{R}})}\approx\|(\sum_{q\in
S_{i}}|E_{q}F|^{2})^{1/2}\|_{L^{p}(B_{\sqrt{R}})}.$

##  4\. Tight decoupling

Let $p\geq 2$ and let $R$ be a large number. To motivate the forthcoming
discussion, we ask how many points $\xi_{1},\ldots,\xi_{N}\in{\mathbb{R}}^{d}$
with $|\xi_{n}|\lesssim 1$ can we find, so that we have $L^{p}$ square root
cancellation on the ball $B(0,R)=\\{x\in{\mathbb{R}}^{d}:\;|x|\leq R\\}$ (or
equivalently, on the square $[0,R]^{d}$), namely

$\frac{1}{|B(0,R)|}\int_{B(0,R)}|\sum_{n=1}^{N}a_{n}e(\xi_{n}\cdot
x)|^{p}dx\lesssim\|a_{n}\|_{l^{2}}^{p}$ (10)

for each $a_{n}\in{\mathbb{C}}$. We ask that the implicit constant in (10) is
independent of $R$. If (10) holds for some $p$, it will also hold for smaller
indices, due to Hölder’s inequality.

It is easy to see that the upper bound $N\lesssim R^{\frac{2d}{p}}$ is
necessary, by considering constant weights $a_{n}=1$. This is due to
constructive interference near the origin

$|\sum_{n=1}^{N}e(\xi_{n}\cdot x)|\sim N,\;\;\text{for }|x|<1/100,$

leading to the inequality

$\frac{N^{p}}{R^{d}}\lesssim N^{p/2}.$

Interestingly, there is a choice (in fact many) of $N\sim R^{2d/p}$ points
$\xi_{n}$ such that (10) holds. This is a consequence of the $\Lambda(p)$
phenomenon, settled by Bourgain in [8].

######  Theorem 4.1.

For each orthonormal system $\varphi_{1},\ldots,\varphi_{n}$ of functions with
$\|\varphi_{i}\|_{L^{\infty}}\leq 1$ and each $p>2$, there is a set
$S\subset\\{1,2,\ldots,n\\}$ with size $\sim n^{2/p}$ such that

$\|\sum_{i\in S}a_{i}\varphi_{i}\|_{p}\lesssim_{p}(\sum_{i\in
S}|a_{i}|^{2})^{1/2}$

for each $a_{i}\in{\mathbb{C}}$. The implicit constant is independent of $n$
and $a_{i}$. A random subset $S$ with this size will work, with high
probability.

Consider the functions $\varphi_{{\bf
n}}(x_{1},\ldots,x_{d})=e({n_{1}x_{1}+\ldots+n_{d}x_{d}})$ with $n_{i}$
positive integers smaller than $R$. They are $L^{2}$ orthogonal on
$[0,1]^{d}$, and there are $\sim R^{d}$ of them. Theorem 4.1 shows the
existence of $\sim(R^{d})^{2/p}$ such functions satisfying

$\int_{[0,1]^{d}}|\sum_{{\bf n}\in S}a_{\bf n}e({\bf n}\cdot
x)|^{p}dx\lesssim\|a_{\bf n}\|_{l^{2}}^{p}.$

Rescaling, it follows that taking $\xi_{n}$ to be the points in $\\{\frac{{\bf
n}}{R}:\;{\bf n}\in S\\}$ satisfies (10).

For generic $S$, the points $\frac{{\bf n}}{R}$ are spread out somewhat
uniformly in the cube $[0,1]^{d}$. We will be interested in the case when the
points lie on a prescribed manifold. Some degree of curvature will be needed,
as the points cannot lie inside any given hyperplane ${\mathbb{H}}$. Indeed,
if $\xi_{n}\in{\mathbb{H}}$ and $a_{n}=1$, we have constructive interference
on the bigger set $E=\\{x\in
B(0,R):\;{\operatorname{dist}\,}(x,{\mathbb{H}}^{\perp})\leq 1/100\\}$ of
volume $\sim R$. This forces the more severe restriction $N\lesssim
R^{\frac{2(d-1)}{p}}$.

If (10) is satisfied for a collection of $\xi_{n}$ with maximal size $N\sim
R^{2d/p}$, we will refer to (10) as tight decoupling. Slightly weaker versions
may also be considered, where $R^{\epsilon}$ losses are tolerated.

Here is our main theorem in this section. To keep things simpler, we will
restrict attention to the constant coefficient case, which is by far the most
relevant for applications. But see also Question 4.5.

######  Theorem 4.2 (Tight decoupling for frequencies in thin sets).

Let $p\geq 2$ and let $E\subset{\mathbb{R}}^{d}$ be a compact set. Assume that
there is a Borel probability measure $d\sigma$ with
${\operatorname{supp}\,}(d\sigma)\subset E$ such that

$\int_{{\mathbb{R}}^{d}}|\widehat{d\sigma}(x)|^{p}dx<\infty.$ (11)

We also assume that for each $\epsilon$ there is a partition of
${\operatorname{supp}\,}(d\sigma)$ into Borel sets with equal $d\sigma$
measure and diameter less than $\epsilon$.

Then for each $R\gg 1$ there is a collection of points
${\mathcal{P}}_{R}\subset E$ with cardinality $\sim R^{\frac{2d}{p}}$ such
that we have square root cancellation on $B(0,R)$ in $L^{p}$ for
$a_{\xi}\equiv 1$

$\frac{1}{|B(0,R)|}\int_{B(0,R)}|\sum_{\xi\in{\mathcal{P}}_{R}}a_{\xi}e(\xi\cdot
x)|^{p}dx\lesssim\|a_{\xi}\|_{l^{2}}^{p}.$ (12)

Conversely, if such ${\mathcal{P}}_{R}\subset E$ exists for a sequence of
$R=R_{k}\to\infty$, then $E$ must support a probability measure $d\sigma$
satisfying $\eqref{decay}$.

Inequality (11) forces the support of $d\sigma$ to have Hausdorff dimension at
least $2d/p$.

######  Example 4.3 (Hypersurfaces).

Perhaps the most interesting class of examples comes from the case when
$d\sigma$ is supported on a smooth hypersurface $\Sigma$. If we assume
$\Sigma$ has nowhere zero Gaussian curvature, then it is known that the
surface measure $d\sigma$ satisfies (11) for each $p>\frac{2d}{d-1}$, since in
fact $|\widehat{d\sigma}(x)|\lesssim(1+|x|)^{-\frac{d-1}{2}}$. Theorem 4.2
shows that tight decoupling is possible in this range.

On the other hand, the Hausdorff dimension of $\Sigma$ is $d-1$, making it a
Salem set. No measure supported on $\Sigma$ can satisfy (11) with
$p<\frac{2d}{d-1}$, so tight decoupling is impossible in this range.

The case $p=\frac{4d}{d-1}$ is particularly revealing. When the points
$\xi_{n}\in\Sigma$ are $1/\sqrt{R}$-separated, it was proved in [4], as a
consequence of decoupling, that (10) holds for $p\leq\frac{2(d+1)}{d-1}$ for
all $a_{n}$, albeit with mild $R^{\epsilon}$ losses. At this level of
generality, this exponent is best possible. To see this, assume $\Sigma$ is
parametrized by $(x_{1},\ldots,x_{d-1},\Phi(x_{1},\ldots,x_{d-1}))$. Let
$\xi_{\bf n}=(\frac{{\bf n}}{\sqrt{R}},\Phi(\frac{{\bf
n}}{\sqrt{R}}))\in\Sigma$, with ${\bf n}\in{\mathbb{Z}}^{d-1}\cap
B_{d-1}(0,\sqrt{R})$. These points are $1/\sqrt{R}$-separated, and there are
$\sim R^{\frac{d-1}{2}}$ many of them. The new phenomenon is that the
associated exponential sums have constructive interference in many regions of
$B(0,R)$, not just near the origin. More precisely,

$|\sum_{{\bf n}}e(\xi_{\bf n}\cdot x)|\sim R^{\frac{d-1}{2}}$

whenever $|x_{i}-l_{i}\sqrt{R}|\leq\frac{1}{100d}$ for each $1\leq i\leq d-1$
and some integer $|l_{i}|\lesssim\sqrt{R}$, and $|x_{d}|\leq 1/100$. This set
has measure $\sim R^{-\frac{d-1}{2}}$, which can be seen to force the
restriction $p\leq\frac{2(d+1)}{d-1}$ in (10).

As a consequence of Theorem 4.2 with $p=\frac{4d}{d-1}$, we find that each
$\Sigma$ as above supports a collection ${\mathcal{P}}_{R}$ of roughly
$R^{\frac{d-1}{2}}$ points $\xi$, such that we have square root cancellation
in $L^{\frac{4d}{d-1}}$

$\frac{1}{|B(0,R)|}\int_{B(0,R)}|\sum_{\xi\in{\mathcal{P}}_{R}}e(\xi\cdot
x)|^{\frac{4d}{d-1}}dx\lesssim R^{d}.$

There is no guarantee that these points are $1/\sqrt{R}$-separated, but our
proof of Theorem 4.2 will show that this can be arranged to be true for a
significant fraction of all pairs of points.

The examples provided by our proof of Theorem 4.2 are non-deterministic, they
arise via random selections. We are not aware of any deterministic example
that is known to satisfy the case $p=\frac{4d}{d-1}$ of the theorem. However,
we recall two conjectures that fit into this category. This time, we allow
$R^{\epsilon}$ losses.

The first one is about lattice points on the sphere. Consider the typical
eigenfunction of the Laplacian on ${\mathbb{T}}^{d}$, given by

$S_{N,d}({\textbf{x}})=\sum_{{\bf
n}\in\sqrt{N}{\mathbb{S}}^{d-1}\cap{\mathbb{Z}}^{d}}a_{{\bf n}}e({\bf
n}\cdot{\textbf{x}}).$

It has been conjectured, see e.g. [7], that if $d\geq 3$ and
$p\leq\frac{2d}{d-2}$

$\int_{[0,1]^{d}}|S_{N,d}(x)|^{p}dx\lesssim_{\epsilon}N^{\epsilon}\|a_{\bf
n}\|_{l^{2}}^{p}.$ (13)

This inequality, sometimes referred to as the discrete Restriction conjecture
for the sphere, has been verified in [4] in the smaller range
$p\leq\frac{2(d+1)}{d-1}$, that is accessible via decoupling.

Let $d=3$, so $\frac{2d}{d-2}=6$ coincides with $\frac{4d}{d-1}$ . Consider an
integer $N$ for which $\sqrt{N}{\mathbb{S}}^{2}\cap{\mathbb{Z}}^{3}$ contains
$\approx\sqrt{N}$ points (this is the case for infinitely many $N$). The
rescaled collection
$\frac{1}{\sqrt{N}}(\sqrt{N}{\mathbb{S}}^{2}\cap{\mathbb{Z}}^{3})$ is
conjectured to satisfy the tight decoupling (12) with $R=\sqrt{N}$, $p=6$ and
$R^{\epsilon}$ losses.

As a second example, we recall a conjecture from [12]. Consider the set of
lattice points in the thin annulus

${\mathcal{A}}_{R,R^{-1/2}}=\\{{\bf n}\in{\mathbb{Z}}^{2}:|{\bf n}-R|\leq
R^{-1/2}\\}.$

Moreover, we may assume that there are $\approx R^{1/2}$ such points. The
rescaled points $\frac{1}{\sqrt{R}}{\mathcal{A}}_{R,R^{-1/2}}$ do not belong
to the circle ${\mathbb{S}}^{1}$, but rather to a thin neighborhood of it. It
is conjectured in [12] (Conjecture B) that

$\int_{[0,1]^{2}}|\sum_{{\bf n}\in{\mathbb{Z}}^{2}:|{\bf n}-R|\leq
R^{-1/2}}e({\bf n}\cdot x)|^{8}dx\lesssim_{\epsilon}R^{2+\epsilon}.$

Thus, the rescaled points are expected to satisfy the tight decoupling (12)
with $p=8$ (and $R^{\epsilon}$ losses). The arbitrary coefficient version of
this inequality is false, as ${\mathcal{A}}_{R,R^{-1/2}}$ may contain many
points in arithmetic progression.

When $d\geq 4$, (13) provides further candidates for tight decoupling, but the
value of $\frac{2d}{d-2}$ is now smaller than $\frac{4d}{d-1}$. For infinitely
many $N$, the set $\sqrt{N}{\mathbb{S}}^{d-1}\cap{\mathbb{Z}}^{d}$ contains
$\approx N^{\frac{d-2}{2}}$ points. This matches $R^{2d/p}$ with $R=\sqrt{N}$
and $p=\frac{2d}{d-2}$. Interestingly, (13) was verified for $d=4$ and $a_{\bf
n}=1$ in [3], using the Siegel mass formula. We are not aware of any other
deterministic example for which tight decoupling is known to hold for
frequencies localized on a hypersurface. This example gives the heuristics for
why (13) is extremely difficult. Tight decoupling is the highest degree on
square root cancellation, and verifying it is expected to involve high-level
number theory.

######  Remark 4.4.

In [18] (see also Lemma 6 in [16] and the main result in [15]), a probability
measure $\nu$ supported on ${\mathbb{P}}^{1}$ is shown to exist for each
$0<\alpha<1$ such that for each $\xi\in{\operatorname{supp}\,}(\nu)$ and
$0<r\lesssim 1$

$r^{\alpha+\epsilon}\lesssim_{\epsilon}\nu(B(\xi,r))\lesssim_{\epsilon}r^{\alpha-\epsilon}$

and

$|\widehat{\nu}(x)|\lesssim_{\epsilon}(1+|x|)^{-\alpha/2+\epsilon}.$ (14)

We use this with $\alpha=1/2$. Consider
$R^{\frac{1}{2}-\epsilon}\lesssim_{\epsilon}N_{R}\lesssim_{\epsilon}R^{\frac{1}{2}+\epsilon}$
pairwise disjoint arcs $J$ on ${\mathbb{P}}^{1}$, with length $1/R$ and such
that

$R^{-\frac{1}{2}-\epsilon}\lesssim_{\epsilon}\nu(J)\lesssim_{\epsilon}R^{-\frac{1}{2}+\epsilon}.$

Let $\psi_{R}(x)=e^{-(\frac{|x|}{R})^{2}}$, so
$\widehat{\psi_{R}}(\xi)=R^{2}e^{-(R{|\xi|})^{2}}$. Let $c_{J}$ be the center
of $J$. Let $\eta$ be smooth, compactly supported, positive, with
$|\widehat{\eta}(x)|\geq 1$ for $|x|\leq 1$. Note that for each
$\xi\in{\mathbb{R}}^{2}$

$0\leq\sum_{J}\eta(R(\xi-
c_{J}))\lesssim_{\epsilon}R^{-3/2+\epsilon}(d\nu)*\widehat{\psi_{R}}(\xi).$

Since $8$ is an even integer, this implies that, taking Fourier transforms on
both sides

$\|{\frac{1}{R^{2}}\sum_{J}\widehat{\eta}(\frac{x}{R})e(c_{J}\cdot
x)}\|_{L^{8}({\mathbb{R}}^{2})}\lesssim_{\epsilon}R^{-3/2+\epsilon}\|\widehat{\nu}\psi_{R}\|_{L^{8}({\mathbb{R}}^{2})}\lesssim_{\epsilon}R^{-\frac{3}{2}+\epsilon}.$

The last inequality follows from (14). We find that

$\|\sum_{J}e(c_{J}\cdot
x)\|_{L^{8}(B_{R})}\lesssim_{\epsilon}R^{\frac{1}{2}+\epsilon}.$

This is tight decoupling with $R^{\epsilon}$ loss. The fact that $8$ is even
integer is crucial to this argument.

This construction is also non deterministic, as $\nu$ is a random measure. The
frequency points $c_{J}$ belong to a fractal. This is very different from our
main construction, where $\xi_{n}$ are well spaced, with high probability.

We close the discussion about hypersurfaces with the following question, open
even for ${\mathbb{P}}_{1}$.

######  Question 4.5.

Let $\frac{2(d+1)}{d-1}\leq p\leq\frac{4d}{d-1}$. Given a hypersurface
$\Sigma\subset{\mathbb{R}}^{d}$ with positive curvatures (e.g. the sphere),
and $R\gg 1$, is there a set ${\mathcal{P}}_{R}\subset\Sigma$ with $\sim
R^{2p/d}$ points such that

$\frac{1}{|B(0,R)|}\int_{B(0,R)}|\sum_{\xi\in{\mathcal{P}}_{R}}a_{\xi}e(\xi\cdot
x)|^{p}dx\lesssim\|a_{\xi}\|_{l^{2}}^{p}$

holds for arbitrary $a_{\xi}$? How about with $R^{\epsilon}$ losses, when
$p>\frac{2(d+1)}{d-1}$?

The additional restriction $p\geq\frac{2(d+1)}{d-1}$ is forced by lower
dimensional concentration. There is a cap with diameter $\sim R^{-1/2}$ on
$\Sigma$ containing $N\sim R^{\frac{2d}{p}-\frac{d-1}{2}}$ of the points in
${\mathcal{P}}_{R}$. Test the inequality with $a_{\xi}=1$ on this set, and
zero otherwise. The points lie inside a box with dimensions $R^{-1/2}$ ($d-1$
times) and $1/R$ (once). There is constructive interference on the box
centered at the origin with dimensions $R^{1/2}$ ($d-1$ times) and $R$ (once).
Thus

$R^{-d}R^{\frac{d-1}{2}+1}N^{p}\lesssim N^{p/2},$

which means $p\geq\frac{2(d+1)}{d-1}$.

######  Example 4.6 (Curves).

Consider a curve with nonzero torsion in ${\mathbb{R}}^{d}$, such as the
moment curve

$\Gamma^{d}=\\{(t,\ldots,t^{d}):\;0\leq t\leq 1\\}.$

The main result in [13] proves that $\Gamma^{d}$ does not support any measure
with Fourier transform in $L^{p}$ for $p<\frac{d^{2}+d+2}{2}$. The second part
of Theorem 4.2 concludes that it is impossible to achieve tight decoupling in
the range $p<\frac{d^{2}+d+2}{2}$, with frequencies in $\Gamma^{d}$.

On the other hand, it is known ([1]) that the uniform measure on $\Gamma^{d}$
satisfies (11) for $p>\frac{d^{2}+d+2}{2}$. When combined with the first part
of Theorem 4.2, we see that tight decoupling is possible in this range.

Let us specialize to $p=2d^{2}$. Theorem (4.2) shows the existence of roughly
$R^{1/d}$ points on $\Gamma^{d}$ such that (12) holds for this $p$. On the
other hand it was proved in [5] that (12) holds for $p\leq d(d+1)$, arbitrary
coefficients and arbitrary collections ${\mathcal{P}}_{R}$ of points on
$\Gamma^{d}$ that are $1/R^{1/d}$-separated. This range follows from
decoupling and is sharp under just the separation condition, as it can be seen
by considering the rescaled lattice points on $\Gamma^{d}$ corresponding to
$t=n/R^{1/d}$, $1\leq n\leq R^{1/d}$.

We use two classical results. The first one is due to Marcinkiewicz–Zygmund.

######  Theorem 4.7.

Let $X_{1},\ldots,X_{N}$ be independent, mean zero random variables belonging
to $L^{p}$ for some $p\geq 1$. Then

$\int|\sum_{n=1}^{N}X_{n}|^{p}\sim_{p}\int(\sum_{n=1}^{N}|X_{n}|^{2})^{p/2}.$

The second one is usually attributed to Rosenthal, see Lemma 1 in [17]. It
does not require mean zero.

######  Theorem 4.8.

Let $Y_{1},\ldots,Y_{N}$ be independent, random variables belonging to $L^{q}$
for some $q\geq 1$. Then

$\int|\sum_{n=1}^{N}Y_{n}|^{q}\lesssim_{q}\sum_{n=1}^{N}\int|Y_{n}|^{q}+(\sum_{n=1}^{N}\int|Y_{n}|)^{q}.$

Let us now prove the first part in Theorem 4.2.

Consider a partition of ${\operatorname{supp}\,}(d\sigma)$ into sets
$S_{1},\ldots,S_{N}$ with $d\sigma$ measure equal to $1/N$ and diameter
$\lesssim R^{-1-\frac{d}{p}}$. Pick any point $\xi_{n}\in S_{n}$. Then, if
$|x|<R$

$\displaystyle|\frac{1}{N}\sum_{n=1}^{N}e(\xi_{n}\cdot
x)-\widehat{d\sigma}(x)|$
$\displaystyle\leq\sum_{n=1}^{N}|\int_{S_{n}}(e(\xi_{n}\cdot x)-e(\xi\cdot
x))d\sigma(\xi)|$ $\displaystyle\leq\frac{1}{N}\sum_{n=1}^{N}\max_{\xi\in
S_{n}}|e(\xi_{n}\cdot x)-e(\xi\cdot x)|$
$\displaystyle\lesssim\max_{n}\max_{\xi\in S_{n}}R|\xi_{n}-\xi|\lesssim
R^{-d/p}.$

Combining this with (11) it follows that

$\int_{B(0,R)}|\sum_{n=1}^{N}e(\xi_{n}\cdot x)|^{p}dx\lesssim
N^{p}(1+\int_{B(0,R)}|\widehat{d\sigma}(x)|^{p}dx)\lesssim N^{p}.$

Assume $\frac{1}{N}\leq\delta\leq 1$. Consider the $\\{0,1\\}$-valued
independent random variables $f_{n}(\omega)$ with mean $\delta$, defined on
the probability space $\Omega$. We first write for each $x\in B(0,R)$

$\int_{\Omega}|\sum_{n=1}^{N}f_{n}(\omega)e(\xi_{n}\cdot
x)|^{p}d\omega\lesssim\delta^{p}|\sum_{n=1}^{N}e(\xi_{n}\cdot
x)|^{p}+\int_{\Omega}|\sum_{n=1}^{N}(f_{n}(\omega)-\delta)e(\xi_{n}\cdot
x)|^{p}d\omega.$

For the second term, we use Theorem 4.7 with
$X_{n}(\omega)=(f_{n}(\omega)-\delta)e(\xi_{n}\cdot x)$ and then Theorem 4.8
with $q=p/2$ and $Y_{n}(\omega)=|f_{n}(\omega)-\delta|^{2}$

$\displaystyle\int_{\Omega}|\sum_{n=1}^{N}(f_{n}(\omega)-\delta)e(\xi_{n}\cdot
x)|^{p}d\omega$
$\displaystyle\lesssim\int_{\Omega}(\sum_{n=1}^{N}|f_{n}(\omega)-\delta|^{2})^{p/2}d\omega$
$\displaystyle\lesssim\sum_{n=1}^{N}\int_{\Omega}|f_{n}(\omega)-\delta|^{p}d\omega+(\sum_{n=1}^{N}\int_{\Omega}|f_{n}(\omega)-\delta|^{2}d\omega)^{p/2}$
$\displaystyle\lesssim
N(\delta+\delta^{p})+(N(\delta+\delta^{2}))^{p/2}\sim(N\delta)^{p/2}.$

Integrating on $B(0,R)$ we get

$\frac{1}{|B(0,R)|}\int_{\Omega}\int_{B(0,R)}|\sum_{n=1}^{N}f_{n}(\omega)e(\xi_{n}\cdot
x)|^{p}dxd\omega\lesssim(N\delta)^{p/2}+\frac{(N\delta)^{p}}{R^{d}}.$

We select $\delta$ such that $N\delta\sim R^{2d/p}$. Let

$\Omega^{\prime}=\\{\omega\in\Omega:\;|\sum_{n=1}^{N}f_{n}(\omega)-N\delta|\geq\frac{N\delta}{2}\\}.$

Then

$|\Omega^{\prime}|\leq\frac{4\int_{\Omega}|\sum_{n=1}^{N}(f_{n}(\omega)-\delta)|^{2}d\omega}{N^{2}\delta^{2}}=\frac{4N(\delta+\delta^{2})}{N^{2}\delta^{2}}\lesssim\frac{1}{N\delta}.$

Thus $|\Omega\setminus\Omega^{\prime}|\geq 1/2$ if $R$ is large enough. We may
thus pick $\omega\in\Omega\setminus\Omega^{\prime}$ such that

$\frac{1}{|B(0,R)|}\int_{B(0,R)}|\sum_{n=1}^{N}f_{n}(\omega)e(\xi_{n}\cdot
x)|^{p}dx\lesssim(N\delta)^{p/2}.$

It suffices to note that $f_{n}(\omega)=1$ for $\sim N\delta$ values of $n$.
The set ${\mathcal{P}}_{R}$ will consist precisely of those $\xi_{n}$
corresponding to these $n$.

We close with the proof of the second part in Theorem 4.2. For each $R=R_{k}$,
consider the probability measure on $E$

$\mu_{k}=\frac{1}{|{\mathcal{P}}_{R_{k}}|}\sum_{\xi\in{\mathcal{P}}_{R_{k}}}\delta_{\xi}.$

Note that

$\widehat{\mu_{k}}(0)=1$

and

$\int_{B(0,R_{k})}|\widehat{\mu_{k}}(x)|^{p}dx\lesssim 1.$

Consider a weak limit point $d\sigma$ of the sequence. Since
$\widehat{\mu_{k}}(x)\to\widehat{d\sigma}(x)$ for each $x$, we find that
$\sigma$ is a probability measure. Also, the dominated convergence theorem
shows that for each ball $B(0,r)$

$\int_{B(0,r)}|\widehat{d\sigma}(x)|^{p}dx=\lim_{k\to\infty}\int_{B(0,r)}|\widehat{\mu_{k}}(x)|^{p}dx\leq\lim_{k\to\infty}\int_{B(0,R_{k})}|\widehat{\mu_{k}}(x)|^{p}dx\lesssim
1.$

## References

  * [1] Arkhipov, G. I., Chubarikov, V. N. and Karatsuba, A. A. Trigonometric sums in number theory and analysis Translated from the 1987 Russian original, De Gruyter Expositions in Mathematics, 39. Walter de Gruyter GmbH $\&$ Co. KG, Berlin, 2004. x+554 pp.
  * [2] Bennett, J, Carbery, A. and Tao, T. On the multilinear restriction and Kakeya conjectures, Acta Math. 196(2006), no.2, 261-302
  * [3] Bourgain, J. and Demeter, C. New bounds for the discrete Fourier restriction to the sphere in four and five dimensions, International Math Research Notices 2015, no. 11, 3150-3184
  * [4] Bourgain, J. and Demeter, C. The proof of the $l^{2}$ Decoupling Conjecture, Ann. of Math. (2) 182 (2015), no. 1, 351-389
  * [5] Bourgain, J., Demeter, C. and Guth, L. Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633-682
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  * [9] Carbery, A. A Remark on reverse Littlewood-Paley, restriction and Kakeya, arxiv.org/abs/1507.02515
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  * [12] Demeter C. and Germain, P. $L^{2}$ to $L^{p}$ bounds for spectral projectors on the Euclidean two-dimensional torus, on arxiv, to appear in Proc. Edinburgh Math. Soc.
  * [13] Guo, S., Iosevich, A., Zhang, R. and Zorin-Kranich, P. $L^{p}$ Integrability of functions with Fourier support on a smooth space curve, arxiv, to appear in IUMJ
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  * [15] Kaufman, R. Random measures on planar curves Ark. Mat., 14(2):245-250, 1976
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English (Australia) |  Australian Institute of Health and Welfare |  government |  Australian Institute of Health and Welfare |  https://www.aihw.gov.au/reports/australias-welfare/understanding-welfare-and-wellbeing
English (Australia) |  Australian Institute of Health and Welfare |  government |  Australian Institute of Health and Welfare |  https://www.aihw.gov.au/reports/australias-welfare/supporting-people-with-disability
English (Australia) |  Royal Commission into Violence, Abuse, Neglect and Exploitation of People with Disability |  government |  Australian Royal Commission into Violence, Abuse, Neglect and Exploitation of People with Disability |  https://disability.royalcommission.gov.au/system/files/2023-09/A%20brief%20guide%20to%20the%20Final%20Report.pdf
English (Australia) |  Department of Social Services |  government |  Australian Government - Summary report: Consultations on the National Housing and Homelessness Plan |  https://engage.dss.gov.au/wp-content/uploads/2024/01/consultation-summary-report-nhhp˙1.pdf
English (Canada) |  Health Canada |  government |  Government of Canada |  https://www.canada.ca/en/health-canada/services/drugs-health-products/biologics-radiopharmaceuticals-genetic-therapies/legislation-guidelines/assisted-human-reproduction/prohibitions-related-surrogacy.html
Language |  Data Producer Organization |  Organization Type |  Source |  Link
---|---|---|---|---
English (Canada) |  Government of Canada |  government |  Government of Canada |  https://www.canada.ca/en/canadian-heritage/services/rights-lgbti-persons.html
English (UK) |  Parliament, House of Commons |  government |  UK Parliament |  https://commonslibrary.parliament.uk/research-briefings/cbp-9920/
English (UK) |  Department of Health & Social Care |  government |  UK Government - Department of Health & Social Care |  https://www.gov.uk/government/publications/having-a-child-through-surrogacy/the-surrogacy-pathway-surrogacy-and-the-legal-process-for-intended-parents-and-surrogates-in-england-and-wales
English (UK) |  UK Government |  government |  UK Government - Department of Health & Social Care |  https://www.gov.uk/legal-rights-when-using-surrogates-and-donors
English (Singapore) |  Ministry of Social and Family Development, Office of the Director-General of Social Welfare (ODGSW) |  government |  Singapore, Ministry of Social and Family Development - Vulnerable Adults Act |  https://www.msf.gov.sg/what-we-do/odgsw/social-insights/2018-vulnerable-adults-act
English (Singapore) |  Ministry of Social and Family Development, Office of the Director-General of Social Welfare (ODGSW) |  government |  Singapore, Ministry of Social and Family Development |  https://www.msf.gov.sg/media-room/article/Update-on-the-Ministrys-position-on-commercial-for-profit-surrogacy
French (Canada) |  Global Affairs Canada |  government |  Canadian Government |  https://www.international.gc.ca/world-monde/issues˙development-enjeux˙developpement/human˙rights-droits˙homme/rights˙lgbti-droits˙lgbti.aspx?lang=fra
French (Canada) |  Femmes et Égalité des genres Canada (en: Women and Gender Equality Canada) |  government |  Canadian Government - 2ELGBTQI+ federal action plan |  https://femmes-egalite-genres.canada.ca/fr/sois-toi-meme/plan-action-federal-2elgbtqi-plus/plan-action-federal-2elgbtqi-plus-2022.html
Turkish |  Adalet ve Kalkınma Partisi (AK PARTİ) (en: Justice and Development Party (AK Party)) |  political party |  election manifesto of the current government 2023 |  https://www.akparti.org.tr/media/bwlbgkif/tu-rkiye-yu-zy%C4%B1l%C4%B1-ic-in-dog-ru-ad%C4%B1mlar-2023-sec-im-beyannamesi.pdf
Language |  Data Producer Organization |  Organization Type |  Source |  Link
---|---|---|---|---
Turkish |  Adalet ve Kalkınma Partisi (AK PARTİ) (en: Justice and Development Party (AK Party)) |  political party |  election manifesto of the current government 2015 |  https://www.akparti.org.tr/media/fmypruoa/7-haziran-2015-edited.pdf
Turkish |  Aile ve Sosyal Hizmetler Bakanlığı (en: Ministry of Family and Social Services) |  government |  Disability Rights National Action Plan 2023-2025 Ministry of Family and Social Services |  https://www.aile.gov.tr/media/133056/engelli˙haklari˙ulusal˙eylem˙plani˙23-25.pdf
Turkish |  Adalet ve Kalkınma Partisi (AK PARTİ) (en: Justice and Development Party (AK Party)) |  political party |  election manifesto of the current government 2018 |  https://www.akparti.org.tr/media/quhdqtia/24-haziran-2018-cumhurbaskanligi-secimleri-ve-genel-secimler-secim-beyannamesi-sayfalar.pdf
Turkish |  Göç İdaresi Başkanlığı (en: Presidency of Migration Management) |  government |  Republic of Türkiye Ministry of Interior Presidency of Migration Management |  https://www.goc.gov.tr/gigm-mevzuati
Turkish |  Göç İdaresi Başkanlığı (en: Presidency of Migration Management) |  government |  Republic of Türkiye Ministry of Interior Presidency of Migration Management |  https://www.goc.gov.tr/hak-ve-yukumlulukler
Turkish |  Cumhuriyet Halk Partisi (CHP) (en: Republican People’s Party) |  political party |  election manifesto of the main opposition 2015 |  https://chp.org.tr/yayin/2015-kasim-secim-bildirgesi
Turkish |  Adalet ve Kalkınma Partisi (AK PARTİ) (en: Justice and Development Party (AK Party)) |  political party |  election manifesto of the current government 2023 |  https://www.akparti.org.tr/media/3qkcsy0c/tu-rkiye-yu-zy%C4%B1l%C4%B1-ic-in-dog-ru-ad%C4%B1mlar-yar%C4%B1n-deg-il-hemen-s-imdi-2023.pdf
Turkish |  Kaos Gey ve Lezbiyen Kültürel Araştırmalar ve Dayanışma Derneği (KAOS GL) (en: Kaos Gay and Lesbian Cultural Research and Solidarity Association) |  civil society group |  KaosGL Organization - LGBT Rights Handbook |  https://kaosgldernegi.org/images/library/2017adalet-bakanligi-lgbt-haklari-el-kitabi.pdf
Table 8: Data sources and data producers for each sources. Prompt |  Annotation 1 |  Annotation 2 |  Annotation 3
---|---|---|---
Fair wages are a question of fair performance and respect. |  human dignity |  labor justice |  labor justice, social assistance
Housing is a basic need. We will help ensure that sufficient living space is created and the construction and living of the future will be affordable, climate-neutral and barrier-free. |  human dignity |  right to housing |  labor justice, social assistance
In order to protect health, we want to take the entire noise situation into account in the future and will examine the introduction of an overall noise assessment. |  social assistance |  right to health |  labor justice, social assistance
Mayors are calling for a change of course because they can no longer meet their obligations to protect those who are under 18 and arrive in Italy without families. |  legal compliance, national security |  integration |  restrictive right to asylum
In Rome, unaccompanied foreign minors spend days in police stations, sometimes with covid. Municipalities sound the alarm: ”There are no more places.” |  legal compliance, national security |  restrictive right to asylum |  integration
Raise awareness among all medical professions about non-discrimination of people living with HIV by health professionals. |  health support, anti-discrimination, sexual equality |  health support |  anti-discrimination
The challenge today is to better take into account the diversity of families in daily life in order to ensure effective equality of rights between all families. |  anti-discrimination, gender inclusivity, sexual equality |  sexuality equality |  anti-discrimination
Authorization to perform part-time service is automatically granted to disabled personnel, after advice from the prevention doctor. |  support, accessibility |  support |  equality
Table 9: Example prompts that necessitated additional discussion by the
authors to resolve annotator disagreement.