# Dynamical localization for polynomial long-range hopping random operators on
$\mathbb{Z}^{d}$

Wenwen Jian  College of Arts and Sciences, Shanghai Polytechnic University,
Shanghai 201209, P. R. China<EMAIL_ADDRESS>and  Yingte Sun  School of
Mathematical Sciences, Yangzhou University, Yangzhou 225009, P. R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove a power-law version dynamical localization for a
random operator $\mathrm{H}_{\omega}$ on $\mathbb{Z}^{d}$ with long-range
hopping. In breif, for the linear Schrödinger equation

$\mathrm{i}\partial_{t}u=\mathrm{H}_{\omega}u,\quad
u\in\ell^{2}(\mathbb{Z}^{d}),$

the Sobolev norm of the solution with well localized initial state is bounded
for any $t\geq 0$.

##  1\. Introduction and the main result

From the breaking working of Anderson [4], a great deal of attention has paid
to the Anderson model (a linear random Schrödinger operator)
$\mathrm{H}_{\omega}$ on $\mathbb{Z}^{d}$, where

$\mathrm{H}_{\omega}=\mathrm{H}_{0}+\mathrm{V}_{\omega}.$

The operator $\mathrm{H}_{0}$ is a negative discrete Laplacian:

$(\mathrm{H}_{0}u)(n)=-\sum_{m\in\mathbb{Z}^{d}:\
\sum^{d}_{v=1}|m_{v}-n_{v}|=1}(u(m)-u(n)).$

The potential $\mathrm{V}_{\omega}$ is a multiplication operator with a
function $\mathrm{V}_{\omega}(n)$ on $\mathbb{Z}^{d}$, that
$\mathrm{V}_{\omega}(n)$ are independent, identically distributed random
variables. We say that the operator $\mathrm{H}_{\omega}$ has exponential
localization, if its spectrum is pure point with exponential decay energy
state. Namely, for some $\alpha>0$, and any energy state $\psi_{k}$, one has

(1.1) $|\psi_{k}(n)|\leq C(k)e^{-\alpha|n|},$

where $C(k)<+\infty$, and depend on energy state.

The mathematicians have development a bit knowledge about the localization of
the random operator $\mathrm{H}_{\omega}$. For $d=1$, they proved that the
exponential localization about the random Schrödinger operator
$\mathrm{H}_{\omega}$ for all energies. For $d\geq 2$, based on the multi-
scale-analysis method(see [8, 13]) and fractional moment method(see [1, 3]),
they can prove that the exponential localization of Anderson model at high
disorder or low energies. However the physicists are more concerned with the
transport properties of the model. In particular, the phenomenon is known as
dynamical localization.

Considering the random operator $\mathrm{H}_{\omega}$ on $\mathbb{Z}^{d}$ with
pure point spectrum, the notion of dynamical localization can be reformulated
as follows: for the Schrödinger equation on $\mathbb{Z}^{d}$,

(1.2) $\mathrm{i}\partial_{t}u=\mathrm{H}_{\omega}u,\quad
u\in\ell^{2}(\mathbb{Z}^{d})$

with well localised initial state $u(0)$, the solution of the equation (1.2)
satisfies that

(1.3)
$\sup_{t}\|u(t)\|_{\mathrm{H}^{s}}=\sup_{t}(\sum_{n\in\mathbb{Z}^{d}}(1+|n|)^{2s}|u_{n}(t)|^{2})^{\frac{1}{2}}<+\infty,$

for any $s>0$. Hence, the dynamical localization is equivalent to the Sobolev
norm of the solution is bounded for the all time.

The first rigorous proof of dynamical localization is attributed to
Aizenmman[2] by employing the fractional moment method. From the point view of
multi-scale analysis technique, an effective way to obtain dynamical
localization is to control the constant $C(k)$ in (1.1). In [7], the authors
introduced the SULE condtition. Namely, all the energy state of random
Schrödinger operator $\mathrm{H}_{\omega}$ have the form of

(1.4) $|\psi_{k}(n)|\leq
D(\epsilon,\omega)e^{\epsilon|n_{k}|}e^{-\alpha|n-n_{k}|},$

where $\epsilon>0$, $D(\epsilon,\omega)$ is an finite constant that does not
depend on the energy of the state, and $n_{k}$ is the localization center
point where $\psi_{k}(n)$ has its maximum. After that, the results of SULE
condition was applied in [6, 7, 12, 15] to prove dynamical localization of
some concrete models.

The above results focus on the models with exponential localization energy
state. However, there are no results involving the models with power-law
localization energy state. Namely, for some $\alpha>0$, and any energy state
$\psi_{k}$, one has

(1.5) $|\psi_{k}(n)|\leq C(k)|n|^{-\alpha},$

where $C(k)$ is finite constant depend on energy state. In [16], Shi study the
$d$ dimensional random operators with long-range hopping, that is

(1.6)
$\mathrm{H}_{\omega}=\lambda^{-1}\mathcal{T}+V_{\omega}(n)\delta_{nn^{\prime}},\quad\lambda\geq
1,$

where $\lambda$ is the coupling constant describing the effect of disorder.
Thorough the multi-scale analysis method, Shi proves that the energy state of
random operator (1.6) exhibits power-law localization of energy state. In this
paper, we try to extend the SULE condition to the power-law localization
energy state and prove a new version of dynamical localization for the random
operator (1.6).

Here, we make some set-up for our main results.

$\star$ The polynomial long-range hopping operator $\mathcal{T}$ is

(1.7) $\mathcal{T}(m,n)=\left\\{\begin{aligned} |m-n|^{-r},&\ \ \ \ \text{for
}m\neq n\text{ with }m,n\in\mathbb{Z}^{d},\\\ 0,\ \ \ \ \ \ &\ \ \ \ \text{for
}m=n\in\mathbb{Z}^{d},\end{aligned}\right.$

where $|n|=\max\limits_{1\leq i\leq d}|n_{i}|$ and $r>0$. From [9], we known
that the operator $\mathcal{T}$ can be viewed as a negative discrete
fractional Laplacian on $\mathbb{Z}^{d}$.

$\star$ $\\{V_{\omega}(n)\\}_{n\in\mathbb{Z}^{d}}$ is independent identically
distributed (_i.i.d._) random variables (with common probability distribution
$\mu$) on some probability space $(\Omega,\mathcal{F},\mathbb{P})$
($\mathcal{F}$ a $\sigma$-algebra on $\Omega$ and $\mathbb{P}$ a probability
measure on $(\Omega,\mathcal{F})$).

Throughout this paper we assume that:

  * •

We have that 111 By Schur’s test and self-adjointness of $\mathcal{T}$, we get
(for $r>d$)
$\displaystyle\|\mathcal{T}\|\leq\sup_{m\in{\mathbb{Z}}^{d}}\sum_{n\neq
m}|m-n|^{-r}\leq\sum_{n\in{\mathbb{Z}}^{d}\setminus\\{0\\}}|n|^{-r}<\infty,$
where $\|\cdot\|$ is the standard operator norm on
$\ell^{2}({\mathbb{Z}}^{d})$.  $d<r<\infty$.

  * •

Let supp$(\mu)=\\{x:\mu(x-\epsilon,x+\epsilon)>0\text{ for any }\epsilon>0\\}$
be the support of the common distribution $\mu$. We assume that supp$(\mu)$
contains at least two points and supp $(\mu)$ is compact:

$\text{supp}(\mu)\subset[-M,M],\ \ \ \ 0<M<\infty.$

Under above assumptions, $\mathrm{H}_{\omega}$ is a bounded self-adjoint
operator on $\ell^{2}(\mathbb{Z}^{d})$ for each $\omega\in\Omega$. Denote by
$\sigma(\mathrm{H}_{\omega})$ the spectrum of $\mathrm{H}_{\omega}$. A well-
known result due to Pastur [14] imply that there exists a set $\Sigma$
(compact and non-random) such that for $\mathbb{P}$ almost all $\omega$,
$\sigma(\mathrm{H}_{\omega})=\Sigma$.

Let us recall the Hölder continuity of a distribution defined in [5].

######  Definition 1.1 ([5]).

We will say a probability measure $\mu$ is Hölder continuous of order $\rho>0$
if

$\displaystyle\frac{1}{\mathcal{K}_{\rho}(\mu)}=\inf_{\kappa>0}\sup_{0<|a-b|\leq\kappa}|a-b|^{-\rho}\mu([a,b])<\infty.$

In this case will call $\mathcal{K}_{\rho}(\mu)>0$ the disorder of $\mu$.

The main result of this paper is the following theorem.

######  Theorem 1.2.

Let ${\mathrm{H}}_{\omega}$ be defined by $(\ref{h})$ with the common
distribution $\mu$ being Hölder continuous of order $\rho>0$, i.e.,
$\mathcal{K}_{\rho}(\mu)>0$. Assume $r\geq\max\\{\frac{200d}{\rho}+25d,\
1800d\\}$. Fix any $0<\kappa<\mathcal{K}_{\rho}(\mu)$. Then there exists
$\lambda_{0}=\lambda_{0}(\kappa,\mu,\rho,{M},r,d)>0$ such that for
$\lambda\geq\lambda_{0}$ and for $\mathbb{P}$ almost all $\omega\in\Omega$,
there exists a positive constant $q:=q(r)\leq\frac{r}{1600}$, such that for
any $\phi\in\ell^{2}(\mathbb{Z}^{d})$ satisfying $|\phi(n)|\leq
C_{\phi}|n|^{-\theta}$, where $\theta\geq r/160$, there exists a constant
$C_{\phi}=C_{\phi}(d,r,q,\theta)$ such that

(1.8) $\||X|^{q/2}e^{-\mathrm{i}\mathrm{H}_{\omega}t}\phi\|^{2}\leq C_{\phi},\
\quad\quad\forall t\geq 0,$

where $X$ is the usual position operator.

######  Remark 1.3.

  *   * •

The core of Theorem 1.2 is to obtain a power-law version SULE condition, that
is, all energy state of random operator (1.6) have the form of

$|\psi_{k,\omega}(n)|\leq
D(\epsilon,\omega)|n_{k}|^{\epsilon}|n-n_{k}|^{-\alpha},$

where $\epsilon>0$ and $D(\epsilon,\omega)$ is a finite constant that does not
depend on energy state.

  * •

It should emphasized that the index $q$ in Theorem 1.2 is closely related to
the index $r$ of operator $\mathcal{T}$. This is different from the case where
exponential localization leads to dynamical localization, where there is no
restriction on the index $q$.

##  2\. Preliminary Knowledge

###  2.1. Sobolev Norm of a Matrix

Let $X_{1},X_{2}\subset\mathbb{Z}^{d}$ be finite sets. Define

$\textbf{M}^{X_{1}}_{X_{2}}=\\{\mathcal{M}=(\mathcal{M}(k,k^{\prime})\in\mathbb{C})_{k\in
X_{1},k^{\prime}\in X_{2}}\\}$

to be the set of all complex matrices with row indexes in $X_{1}$ and column
indexes in $X_{2}$. If $Y_{1}\subset X_{1}$, $Y_{2}\subset X_{2}$, we write
$\mathcal{M}^{Y_{1}}_{Y_{2}}=(\mathcal{M}(k,k^{\prime}))_{k\in
Y_{1},k^{\prime}\in Y_{2}}$ for any
$\mathcal{M}\in\mathbf{M}^{X_{1}}_{X_{2}}$.

######  Definition 2.1.

Let $\mathcal{M}\in\mathbf{M}^{X_{1}}_{X_{2}}$. Define for $s\geq s_{0}$ the
Sobolev norms of $\mathcal{M}$ as:

$\|\mathcal{M}\|_{s}^{2}=C_{0}(s_{0})\sum_{k\in
X_{1}-X_{2}}\left(\sup_{k_{1}-k_{2}=k}|\mathcal{M}(k_{1},k_{2})|\right)^{2}\langle
k\rangle^{2s},$

where $\langle k\rangle=\max\\{1,|k|\\}$ and $C_{0}(s_{0})>0$ is a constant
depending on $s_{0}$.

###  2.2. Green’s Function Estimate

The Green’s function plays a key role in spectral theroy. In this subsection
we present the first main result about Green’s function estimate. For
$n\in{\mathbb{Z}}^{d}$ and $L>0$, define the cube
$\Lambda_{L}(n)=\\{k\in{\mathbb{Z}}^{d}:\ |k-n|\leq L\\}$. Moreover, write
$\Lambda_{L}=\Lambda_{L}(0)$. The volume of a finite set
$\Lambda\subset{\mathbb{Z}}^{d}$ is defined to be $|\Lambda|=\\#\Lambda$. We
have $|\Lambda_{L}(n)|=(2L+1)^{d}$ ($L\in\mathbb{N}$) for example.

If $\Lambda\subset\mathbb{Z}^{d}$, denote
${\mathrm{H}}_{\Lambda}=R_{\Lambda}{\mathrm{H}_{\omega}}R_{\Lambda}$, where
$R_{\Lambda}$ is the restriction operator. Define the Green’s function (if it
exists) as

$\displaystyle G_{\Lambda}(E)=({\mathrm{H}}_{\Lambda}-E)^{-1},\
E\in\mathbb{R}.$

Let us introduce good cubes in ${\mathbb{Z}}^{d}$.

######  Definition 2.2.

Fix $\tau^{\prime}>0$, $\delta\in(0,1)$ and $d/2<s_{0}\leq r_{1}<r-d/2$. We
call $\Lambda_{L}(n)$ is $(E,\delta)$-good if $G_{\Lambda_{L}(n)}(E)$ exists
and satisfies

$\displaystyle\|G_{\Lambda_{L}(n)}(E)\|_{s}\leq L^{\tau^{\prime}+\delta{s}}\
\mathrm{for}\ \forall\ s\in[s_{0},r_{1}].$

Otherwise, we call $\Lambda_{L}(n)$ is $(E,\delta)$-bad. We call
$\Lambda_{L}(n)$ an $(E,\delta)$-good (resp. $(E,\delta)$-bad) $L$-cube if it
is $(E,\delta)$-good (resp. $(E,\delta)$-bad).

######  Remark 2.3.

Let $\zeta\in(\delta,1)$ and $\tau^{\prime}-(\zeta-\delta)r_{1}<0$. Suppose
that $\Lambda_{L}(n)$ is $(E,\delta)$-good. Then we have for $L\geq
L_{0}(\zeta,\tau^{\prime},\delta,r_{1},d)>0$ and
$|n^{\prime}-n^{\prime\prime}|\geq L/2$,

(2.1) $\displaystyle|G_{\Lambda_{L}(n)}(E)(n^{\prime},n^{\prime\prime})|$
$\displaystyle\leq|n^{\prime}-n^{\prime\prime}|^{-(1-\zeta)r_{1}}.$

Assume the following relations hold true:

(2.2) $\displaystyle\left\\{\begin{aligned}
&-(1-\delta)r_{1}+\tau^{\prime}+2s_{0}<0,\\\ &-\xi
r_{1}+\tau^{\prime}+\alpha\tau+(3+\delta+4\xi)s_{0}<0,\\\
&\alpha^{-1}(2\tau^{\prime}+2\alpha\tau+(5+4\xi+2\delta)s_{0})+s_{0}<\tau^{\prime},\\\
\end{aligned}\right.$

where $\alpha,\tau,\tau^{\prime},r_{1}>1,\xi>0,s_{0}>d/2$ and
$\delta\in(0,1)$.

######  Definition 2.4.

We call a site $n\in\Lambda\subset{\mathbb{Z}}^{d}$ is $(l,E,\delta)$-good
with respect to (w.r.t) $\Lambda$ if there exists some
$\Lambda_{l}(m)\subset\Lambda$ such that $\Lambda_{l}(m)$ is $(E,\delta)$-good
and $n\in\Lambda_{l}(m)$ with ${\rm
dist}(n,\Lambda\setminus\Lambda_{l}(m))\geq l/2$. Otherwise, we call
$n\in\Lambda\subset{\mathbb{Z}}^{d}$ is $(l,E,\delta)$-bad w.r.t $\Lambda$.

Let

(2.3) $\displaystyle\tau>(2p+(2+\rho)d)/\rho.$

The multi-scale analysis argument on Green’s function estimate is

######  Theorem 2.5 (Theorem 4.4 in [16]).

Let $\mu$ be Hölder continuous of order $\rho>0$ (i.e.,
$\mathcal{K}_{\rho}(\mu)>0$). Fix $E_{0}\in\mathbb{R}$ with $|E_{0}|\leq
2(\|\mathcal{T}\|+M)$, and assume (2.2), (2.3) hold true. Assume further that

(2.4) $(1+\xi)/\alpha\leq\delta,\quad p>\alpha d+2\alpha p/J$

for $J\in 2\mathbb{N}$. Then for $0<\kappa<\mathcal{K}_{\rho}(\mu)$, there
exists

$\underline{L}_{0}=\underline{L}_{0}(\kappa,\mu,\rho,\|\mathcal{T}\|_{r_{1}},M,J,\alpha,\tau,\xi,\tau^{\prime},\delta,p,{r_{1}},s_{0},d)>0$

such that the following holds: For $L_{0}\geq\underline{L}_{0}$, there is some
$\lambda_{0}=\lambda_{0}(L_{0},\kappa,\rho,p,s_{0},d)>0$ and
$\eta=\eta(L_{0},\kappa,\rho,p,d)>0$ so that for $\lambda\geq\lambda_{0}$ and
$k\geq 0$, we have

$\displaystyle\mathbb{P}({\ \exists}\ E\in[E_{0}-\eta,E_{0}+\eta],\ {\rm s.t.\
}\ \Lambda_{L_{k}}(m)\ {\rm and}\ \Lambda_{L_{k}}(n)\ {\rm are\
}(E,\delta)\text{-{\rm{bad}}})\leq L_{k}^{-2p}$

for all $|m-n|>2L_{k}$, where $L_{k+1}=[L_{k}^{\alpha}]$ and
$L_{0}\geq\underline{L}_{0}$.

###  2.3. Power-law localization

Recall the Poisson’s identity: Let
$\psi=\\{\psi(n)\\}\in{\mathbb{C}}^{\mathbb{Z}^{d}}$ satisfy
$\mathrm{H}_{\omega}\psi=E\psi$. Assume further $G_{\Lambda}(E)$ exists for
some $\Lambda\subset{\mathbb{Z}}^{d}$. Then for any $n\in\Lambda$, we have

(2.5)
$\displaystyle\psi(n)=-\sum_{n^{\prime}\in\Lambda,n^{\prime\prime}\notin\Lambda}\lambda^{-1}G_{\Lambda}(E)(n,n^{\prime})\mathcal{T}(n^{\prime},n^{\prime\prime})\psi(n^{\prime\prime}).$

From Shnol’s Theorem of [11] in long-range operator case, to prove pure point
spectrum of $\mathrm{H}_{\omega}$, it suffices to show that each
$\varepsilon$-generalized eigenfunction belongs to
$\ell^{2}({\mathbb{Z}}^{d})$. In [16], Shi shows that every
$\varepsilon$-generalized (with $0<\varepsilon\leq c(d)\ll 1$) eigenfunction
$\psi$ of random operator (1.6) decays as $|\psi(n)|\leq{|n|^{-{r}/{600}}}$
for $|n|\gg 1$. Specifically, Shi obtains polynomially decaying of each
generalized eigenfunction of $\mathrm{H}_{\omega}$ for $\mathbb{P}$ a.e.
$\omega$. This yields the power-law localization:

######  Theorem 2.6 (Theorem 2.5 in [16]).

Let ${\mathrm{H}}_{\omega}$ be defined by $(\ref{h})$ with the common
distribution $\mu$ being Hölder continuous of order $\rho>0$, i.e.,
$\mathcal{K}_{\rho}(\mu)>0$. Let $r\geq\max\\{\frac{100d+23\rho
d}{\rho},331d\\}$. Fix any $0<\kappa<\mathcal{K}_{\rho}(\mu)$. Then there
exists $\lambda_{0}=\lambda_{0}(\kappa,\mu,\rho,{M},r,d)>0$ such that for
$\lambda\geq\lambda_{0}$, ${\mathrm{H}}_{\omega}$ has pure point spectrum for
$\mathbb{P}$ almost all $\omega\in\Omega$. Moreover, for $\mathbb{P}$ almost
all $\omega\in\Omega$, there exists a complete system of eigenfunctions
$\psi_{j,\omega}=\\{\psi_{j,\omega}(n)\\}_{n\in{\mathbb{Z}}^{d}},\
j=1,2,\cdots,$ satisfying

(2.6) $|\psi_{j,\omega}(n)|\leq C_{j,\omega}|n|^{-r/600},\quad\quad|n|\gg 1.$

######  Remark 2.7.

Note that the coefficients $C_{j,\omega}$ in (2.6) depend on the selection of
energy state.

######  Lemma 2.8 (Lemma A.1 in [16]).

Let $L>2$ with $L\in\mathbb{N}$ and $\Theta-d>1$. Then we have that

(2.7) $\displaystyle\sum_{n\in{\mathbb{Z}}^{d}:\ |n|\geq L}|n|^{-\Theta}\leq
C(\Theta,d)L^{-(\Theta-d)/2},$

where $C(\Theta,d)>0$ depends only on $\Theta,d$.

##  3\. Proof of Theorem 1.2

In order to obtain dynamical localization, it’s need to control the location
and the size of the boxes outside of which the eigenfunctions has “effective”
decrease.

######  Theorem 3.1.

For the operator $\mathrm{H}_{\omega}$ defined by (1.6), we assume that
$\kappa,\lambda_{0}$ and $\psi_{j,\omega},\ j\in\mathbb{N}$ satisfy Theorem
2.6. Let $r\geq\max\\{\frac{200d}{\rho}+25d,331d\\}$. Then for
$\lambda\geq\lambda_{0}$, there exists centers $n_{j,\omega}$ associated to
the eigenfunctions $\psi_{j,\omega}$ with eigenvalues $E_{j,\omega}$ such that
for any $\gamma\in[0,\frac{r}{160}]$ and any
$\epsilon^{\prime}\in[\frac{1}{3},\frac{1}{2})$, there exists a constant
$C(\epsilon^{\prime},\gamma)>0$ such that

(3.1) $|\psi_{j,\omega}(n)|\leq
C(\epsilon^{\prime},\gamma)|n_{j,\omega}|^{\epsilon^{\prime}\gamma}|n-n_{j,\omega}|^{-\gamma},\quad\quad\forall
n\in\mathbb{Z}^{d}.$

Here, $C(\epsilon^{\prime},\gamma)$ dose not depend on $j$(the eigenfunction).

From Theorem 3.1, the eigenfunctions $\psi_{j,\omega}$ are localized outside
boxes of size $|n_{j,\omega}|/2$ around “centers” $n_{j,\omega}$. This result
is stronger than the power-law localization.

In the following, we choose appropriate parameters satisfying Remark 2.3,
(2.2)-(2.4). For this purpose, we can set by direct calculation that

(3.2)
$\displaystyle\alpha=6,\quad\delta=\frac{1}{2},\quad\xi=2,\quad\zeta=\frac{19}{20},\quad
p=13d.$

We define $J_{\star}=J_{\star}(d,\varepsilon)$ to be the smallest even integer
satisfying $p>6d+\frac{12}{J_{\star}}p$. As a consequence, we can set

(3.3) $\displaystyle\tau=\frac{29d}{\rho}+d,\quad
s_{0}=\frac{3}{4}d,\quad\tau^{\prime}=\frac{87d}{\rho}+7d,$

(3.4) $r\geq\max\left\\{\frac{200d}{\rho}+25d,\ 331d\right\\},\qquad
r_{1}=r-8d.$

In order to prove Theorem 1.2, we need the following lemma which says that if
$\psi$ is an eigenfunction of $\mathrm{H}_{\omega}$ with eigenvalue $E$, then
$E$ must be close to the spectrum of $\mathrm{H}_{\Lambda_{L}(n)}$ provided
$L$ is big enough and $\Lambda_{L}(n)$ is centered on a maximum of
$|\psi(n)|$.

######  Lemma 3.2.

There exists a constant $L_{\star}(d,r)$ so that if
$\psi\in\ell^{2}(\mathbb{Z}^{d})$ is an eigenfunction of $\mathrm{H}_{\omega}$
with eigenvalue $E$, and $n_{\star}$ satisfies
$|\psi(n_{\star})|=\sup\\{|\psi(n)|,\ n\in\mathbb{Z}^{d}\\}$, then
$\Lambda_{L}(n_{\star})$ is $(E,1/2)$-bad for all $L\geq L_{\star}(d,r)$.

###### Proof.

Let $\psi\in\ell^{2}(\mathbb{Z}^{d})$ be as in the lemma, so $n_{\star}$
exists. Suppose that $\Lambda_{L_{k}}(n_{\star})$ is $(E,1/2)$-good for some
$k\geq k_{1}=k_{1}(d,r)$ sufficiently large. Applying the identity (2.5) at
the point $n_{\star}$, one has that

$\displaystyle|\psi(n_{\star})|$
$\displaystyle\leq\sum_{n^{\prime}\in\Lambda_{L_{k}}(n_{\star}),\atop
n^{\prime\prime}\notin\Lambda_{L_{k}}(n_{\star})}C(d)|G_{\Lambda_{L_{k}}(n_{\star})}(E)(n_{\star},n^{\prime})|\cdot|n^{\prime}-n^{\prime\prime}|^{-r}|\psi(n^{\prime\prime})|$
$\displaystyle\leq\ {\rm(V)}+{\rm(VI)},$

where

$\displaystyle{\rm(V)}$ $\displaystyle=\sum_{|n^{\prime}-n_{\star}|\leq
L_{k}/2,\atop|n^{\prime\prime}-n_{\star}|>L_{k}}C(d,s_{0})L_{k}^{\tau^{\prime}+\frac{1}{2}s_{0}}|n^{\prime}-n^{\prime\prime}|^{-r}|\psi(n_{\star})|,$
$\displaystyle{\rm(VI)}$
$\displaystyle=\sum_{L_{k}/2<|n^{\prime}-n_{\star}|\leq
L_{k},\atop|n^{\prime\prime}-n_{\star}|>{L_{k}}}C(d)|n^{\prime}-n_{\star}|^{-\frac{r_{1}}{20}}|n^{\prime}-n^{\prime\prime}|^{-r}|\psi(n_{\star})|.$

When $|n^{\prime}-n_{\star}|\leq L_{k}/2$ and
$|n^{\prime\prime}-n_{\star}|>L_{k}$, one has that
$|n^{\prime}-n^{\prime\prime}|\geq|n^{\prime\prime}-n_{\star}|-|n^{\prime}-n_{\star}|>|n^{\prime\prime}-n_{\star}|/2$.
By (2.7), we have that

$\displaystyle{\rm(V)}$
$\displaystyle\leq\sum_{|n^{\prime\prime}-n_{\star}|>L_{k}}C(r,d,s_{0})L_{k}^{d+\tau^{\prime}+\frac{1}{2}s_{0}}|n^{\prime\prime}-n_{\star}|^{-r}|\psi(n_{\star})|$
$\displaystyle\leq
C(d,r,s_{0})L_{k}^{-\frac{r}{2}+\tau^{\prime}+\frac{1}{2}s_{0}+\frac{3}{2}d}|\psi(n_{\star})|,$

For the term (VI), one has that

$\displaystyle{\rm(VI)}=$ $\displaystyle\
(\sum_{L_{k}/2<|n^{\prime}-n_{\star}|\leq L_{k},\atop
L_{k}<|n^{\prime\prime}-n_{\star}|<2L_{k}}+\sum_{L_{k}/2<|n^{\prime}-n_{\star}|\leq
L_{k},\atop|n^{\prime\prime}-n_{\star}|\geq{2L_{k}}})C(d)|n^{\prime}-n_{\star}|^{-\frac{r_{1}}{20}}|n^{\prime}-n^{\prime\prime}|^{-r}|\psi(n_{\star})|$
$\displaystyle\leq$ $\displaystyle\ \sum_{L_{k}/2<|n^{\prime}-n_{\star}|\leq
L_{k}}C(d,r)L_{k}^{d}|n^{\prime}-n_{\star}|^{-\frac{r_{1}}{20}}|\psi(n_{\star})|$
$\displaystyle\ \ +\sum_{L_{k}/2<|n^{\prime}-n_{\star}|\leq
L_{k},\atop|n^{\prime\prime}-n_{\star}|\geq{2L_{k}}}C(d,r)|n^{\prime}-n_{\star}|^{-\frac{r_{1}}{20}}|n^{\prime\prime}-n_{\star}|^{-r}|\psi(n_{\star})|$
$\displaystyle\leq$ $\displaystyle\
C(d,r_{1})L_{k}^{-\frac{r_{1}}{20}+2d}|\psi(n_{\star})|+C(d,r_{1},r)L_{k}^{-\frac{r_{1}}{20}-\frac{r}{2}+\frac{3}{2}d}|\psi(n_{\star})|$
$\displaystyle\leq$ $\displaystyle\
C(d,r_{1},r)L_{k}^{-\frac{r_{1}}{20}+2d}|\psi(n_{\star})|.$

Recalling (3.2)-(3.4), one has that

$-\frac{r}{2}+\tau^{\prime}+\frac{1}{2}s_{0}+\frac{3}{2}d<-\frac{r}{25},\quad-\frac{r_{1}}{20}+2d<-\frac{r}{25}.$

Hence, for large enough $k$ (depending on $d$ and $r$),

$|\psi(n_{\star})|\leq
L_{k}^{-\frac{r}{30}}|\psi(n_{\star})|<|\psi(n_{\star})|,$

which is impossible. Therefore $\Lambda_{L_{k}}(n_{\star})$ is $(E,1/2)$-bad
for all $k\geq k_{1}(d,r)$. The lemma is proved. ∎

Now we can give the proof of Theorem 3.1.

###### Proof of Theorem 3.1.

Under the hypotheses of Theorem 3.1 and Theorem 2.6, $\mathrm{H}_{\omega}$ has
power-law localization for $\mathbb{P}$ almost all $\omega\in\Omega$. This
means that there exists $\Omega_{1}\subset\Omega$, $\mu(\Omega_{1})=1$ so that
for all $\omega\in\Omega_{1}$, $\sigma_{c}(\mathrm{H}_{\omega})=\emptyset$ and
for all eigenvalues $E_{j,\omega}$, the corresponding eigenfunction
$\psi_{j,\omega}$ is $\ell^{2}$ and satisfies (2.6) with
$\|\psi_{j,\omega}\|=1$.

Our ultimate goal is to control the constant $C_{j,\omega}$ in (2.6). More
precisely, for each $0<\gamma\leq r/400$, we try to show that $n_{j,\omega}$
can be chosen so that $C_{j,\omega}$ grows slower than
$|n_{j,\omega}|^{\epsilon^{\prime}\gamma}$ for $1/3\leq\epsilon^{\prime}<1/2$.

The outline of the proof is a little complicated, for the convenience of
reader, we will divide the main proof into three parts. Firstly, we fix
$L_{0}=\underline{L}_{0}$, $\lambda_{0},\ \eta$ and
$I=[E_{0}-\eta,E_{0}+\eta]$ in Theorem 2.5. Recalling Theorem 2.5, we have for
$\lambda\geq\lambda_{0}$ and $k\geq 0$,

$\displaystyle\mathbb{P}({\ \exists}\ E\in I,\ {\rm s.t.\ both}\
\Lambda_{L_{k}}(m)\ {\rm and}\ \Lambda_{L_{k}}(n)\ {\rm are\
}(E,\delta)\text{-{\rm{bad}}})\leq L_{k}^{-2p}$

for all $|m-n|>2L_{k}$, where $L_{k+1}=[L_{k}^{\alpha}]$ and $L_{0}\gg 1$.

Step one. For any $k\geq 0$, we define the set

$A_{k+1}(n_{0})=\Lambda_{2L_{k+1}}(n_{0})\setminus\Lambda_{L_{k}}(n_{0})$

and the event:

(3.5) ${\mathbf{E}}_{k}(n_{0})=\\{\exists E,\ \exists n\in
A_{k+1}(n_{0}),\text{ s.t.\ both\ }\Lambda_{L_{k}}(n_{0})\text{ and
}\Lambda_{L_{k}}(n)\text{ are }(E,1/2)\text{-{bad}}\\}.$

From Theorem 2.5,

$\mathbb{P}({\mathbf{E}}_{k}(n_{0}))\leq\sum_{n\in
A_{k+1}(n_{0})}L_{k}^{-2p}\leq C(d)(4L_{k}^{\alpha}+1)^{d}L_{k}^{-2p}\leq
C(d)L_{k}^{-2p+\alpha d}.$

For $1/3\leq\epsilon^{\prime}<1/2$, we define

$F_{k}=\bigcup_{|n_{0}|\leq
L_{k+1}^{1/\epsilon^{\prime}}}\mathbf{E}_{k}(n_{0}).$

Then

$\mathbb{P}(F_{k})\leq\sum_{|n_{0}|\leq
L_{k+1}^{1/\epsilon^{\prime}}}\mathbb{P}(E_{k}(n_{0}))\leq
C(d,\epsilon^{\prime})L_{k}^{-2p+\alpha d(1+\frac{1}{\epsilon^{\prime}})},$

where $p$ and $\alpha$ are defined in (3.2). It is easy to verified that
$\sum_{k=0}^{\infty}\mathbb{P}(F_{k})<\infty$. Then, the Borel-Cantelli lemma
implies that

$\mathbb{P}\left(\lim_{m\rightarrow\infty}\bigcup_{k\geq m}F_{k}\right)=0,$

so that the set

$\Omega_{2}=\\{\omega\in\Omega_{1}:\
\exists\tilde{k}_{1}=\tilde{k}_{1}(p,d,\epsilon^{\prime}),\text{ s.t. }\forall
k\geq\tilde{k}_{1},\ \omega\notin F_{k}\\}$

has full measure.

Now pick $\omega\in\Omega_{1}\cap\Omega_{2}$, which will be kept fixed
throughout the rest of the proof. Let $\psi_{j,\omega}$ be the eigenfunction
of energy $E_{j,\omega}$, and $n_{j,\omega}$ be a point where
$|\psi_{j,\omega}(n_{j,\omega})|$ is maximal. Note that such a point exists,
since $\omega\in\Omega_{1}$ and $\psi_{j,\omega}\in\ell^{2}(\mathbb{Z}^{d})$.
Furthermore, we define the integers

(3.6) $\hat{k}_{2}(\epsilon^{\prime},m)=\min\\{k\geq 0\text{ such that
}|m|^{\epsilon^{\prime}}<L_{k+1}\\},\quad m\in\mathbb{Z}^{d},$

and

(3.7)
$\bar{k}_{2}=\bar{k}_{2}(p,d,\epsilon^{\prime},n_{j,\omega})=\max\\{\tilde{k}_{1},\hat{k}_{2}(\epsilon^{\prime},n_{j,\omega})\\}.$

For any $k\geq\bar{k}_{2}$, we see that $\omega\notin E_{k}(n_{j,\omega})$
from the definition of $F_{k}$. By (3.5), $\forall k\geq\bar{k}_{2}$ and
$\forall n\in A_{k+1}(n_{j,\omega})$, either $\Lambda_{L_{k}}(n_{j,\omega})$
or $\Lambda_{L_{k}}(n)$ is $(E_{j,\omega},1/2)$-good. Applying Lemma 3.2,
there is an integer

$k_{2}=\max\\{k_{1},\bar{k}_{2}\\}=\max\\{k_{1},\tilde{k}_{2},\hat{k}_{2}\\},$

where $k_{1}={k}_{1}(d,r)$ and
$\tilde{k}_{2}:=\tilde{k}_{2}(p,d,r,\epsilon^{\prime})$ does not depend on
$j$, such that for any energy $E_{j,\omega}$,

$\Lambda_{L_{k}}(n)\text{ is }(E_{j,\omega},1/2)\text{-good},\quad\forall n\in
A_{k+1}(n_{j,\omega}),\quad\forall k\geq k_{2}.$

Step two. Let us apply the Possion’s identity (2.5) at the point $n\in
A_{k+1}(n_{j,\omega})$. Similarly to the proof of Lemma 3.2, and recalling
(3.2)-(3.4), for any $k\geq k_{2}$, one has that

$\displaystyle|\psi_{j,\omega}(n)|$
$\displaystyle\leq\sum_{n^{\prime}\in\Lambda_{L_{k}(n)},\atop
n^{\prime\prime}\notin\Lambda_{L_{k}(n)}}|G_{\Lambda_{L_{k}}(n)}(E_{j,\omega})(n,n^{\prime})|\cdot|n^{\prime}-n^{\prime\prime}|^{-r}|\psi_{j,\omega}(n^{\prime\prime})|$
$\displaystyle\leq
C(r,d,s_{0})L_{k}^{-\frac{r}{2}+\tau^{\prime}+\frac{1}{2}s_{0}+\frac{3d}{2}}+C(d,r,r_{1})L_{k}^{-\frac{r_{1}}{20}+2d}$
$\displaystyle\leq C(d,r)L_{k}^{-\frac{r}{25}}.$

Set

(3.8)
$\widetilde{A}_{k+1}(n)=\Lambda_{\frac{8}{5}L_{k+1}}(n)\setminus\Lambda_{\frac{4}{3}L_{k}}(n)\subset
A_{k+1}(n).$

If $n\in\widetilde{A}_{k+1}(n_{j,\omega})$, one has that
$L_{k}\geq(\frac{5}{8}|n-n_{j,\omega}|)^{\frac{1}{6}}$, and

$|\psi_{j,\omega}(n)|\leq C(d,r)|n-n_{j,\omega}|^{-\frac{r}{150}}.$

Hence, one can find $k_{3}=\max\\{\tilde{k}_{3},k_{2}\\}$, where
$\tilde{k}_{3}=\tilde{k}_{3}(d,r)$ is independent of $j$, such that

$|\psi_{j,\omega}(n)|\leq|n-n_{j,\omega}|^{-\frac{r}{160}},\qquad\forall
k\geq{k}_{3}.$

Then for any $0<\gamma\leq r/160$,

$|\psi_{j,\omega}(n)|\leq|n-n_{j,\omega}|^{-\gamma},\quad\forall
n\in\widetilde{A}_{k+1}(n_{j,\omega}),\quad\forall k\geq k_{3}.$

Since $\bigcup\limits_{k\geq
k_{3}}\widetilde{A}_{k+1}(n_{j,\omega})=\\{n\in\mathbb{Z}^{d}:\
|n-n_{j,\omega}|>\frac{4}{3}L_{k_{3}}\\}$, there exists
$k_{4}=\max\\{\tilde{k}_{4},k_{3}\\}$, where $\tilde{k}_{4}$ does not depend
on $j$, such that

$|\psi_{j,\omega}(n)|\leq|n-n_{j,\omega}|^{-\gamma},\qquad\forall
n:|n-n_{j,\omega}|\geq L_{k_{4}}.$

Step three. Using the fact that $|\psi_{j,\omega}(n)|\leq 1$ for all
$n\in\mathbb{Z}^{d}$, one has

(3.9) $|\psi_{j,\omega}(n)|\leq
C(\epsilon^{\prime},\gamma)L_{k_{4}}^{\gamma}|n-n_{j,\omega}|^{-\gamma},\qquad\forall
n\in\mathbb{Z}^{d}.$

Now, we try to control the $j$-dependence of the constant
$L_{k_{4}}^{\gamma}$. Note that the only $j$-dependence of $k_{4}$ comes from
$\hat{k}_{2}(\epsilon^{\prime},n_{j,\omega})$. Suppose
$\sup\\{|n_{j,\omega}|\\}<\infty$, then $k_{4}$ can be chosen
$j$-independently, so that we actually obtain a uniform localization for the
all energy state

$|\psi_{j,\omega}(n)|\leq
C(\epsilon^{\prime},\gamma)|n-n_{j,\omega}|^{-\gamma},\quad\forall
n\in\mathbb{Z}^{d}.$

But the following Lemma 3.3 contradicts this first possibility. So, in fact,
$\sup\\{|n_{j,\omega}|\\}=\infty$, and for $j$ sufficiently large, one has

$k_{4}=\hat{k}_{2}(\epsilon^{\prime},n_{j,\omega}),$

and recalling the definition of $\hat{k}_{2}(\epsilon^{\prime},n_{j,\omega})$
in (3.6),

$L_{k_{4}}\leq|n_{j,\omega}|^{\epsilon^{\prime}}.$

Inserting this in (3.9) yields the announced result. Theorem 3.1 is proved. ∎

We also need to control the growth of $|n_{j,\omega}|$ with $j$, which is
given by the following preliminary lemma:

######  Lemma 3.3.

Assume that $n_{j,\omega}$ are defined in Theorem 3.1. Then one can order
$|n_{j,\omega}|$ in increasing order such that for $j$ larger enough,

$|n_{j,\omega}|\geq C|j|^{\frac{1}{d}}.$

###### Proof.

From Theorem 3.1, $\\{\psi_{j,\omega}:j=1,2,\cdots\\}$ is a complete
normalized orthogonal system of $\ell^{2}(\mathbb{Z}^{d})$ and each
$n_{j,\omega}$ is chosen so that
$|\psi_{j,\omega}(n_{j,\omega})|=\sup\\{|\psi_{j,\omega}(n)|,\
n\in\mathbb{Z}^{d}\\}$. Therefore we have

(3.10) $\ \ \ \ \
\sum_{n\in\mathbb{Z}^{d}}|\psi_{j,\omega}(n)|^{2}=1,\qquad\forall
j=1,2,\cdots,$ (3.11)
$\sum_{j=1}^{\infty}|\psi_{j,\omega}(n)|^{2}=1,\qquad\forall
n\in\mathbb{Z}^{d}.$

Fix $0<\varepsilon<1$. For $j\in\mathbb{N}\setminus\\{0\\}$, by using Theorem
3.1 and (2.7), one has that

$\displaystyle\sum_{|n-n_{j,\omega}|\geq\varepsilon
L}|\psi_{j,\omega}(n)|^{2}$ $\displaystyle\leq
C(\epsilon^{\prime},\gamma)\sum_{|n-n_{j,\omega}|\geq\varepsilon
L}|n_{j,\omega}|^{2\epsilon^{\prime}\gamma}|n-n_{j,\omega}|^{-2\gamma}$
$\displaystyle\leq
C(d,\epsilon^{\prime},\gamma)\varepsilon^{-\gamma+\frac{d}{2}}|n_{j,\omega}|^{2\epsilon^{\prime}\gamma}L^{-\gamma+\frac{d}{2}}.$

Assume $|n_{j,\omega}|\leq L$. Then

$\sum_{|n|\geq(1+\varepsilon)L}|\psi_{j,\omega}(n)|^{2}\leq\sum_{|n-n_{j,\omega}|\geq\varepsilon
L}|\psi_{j,\omega}(n)|^{2}\leq
C(d,\epsilon^{\prime},\gamma)\varepsilon^{-\gamma+\frac{d}{2}}L^{-\gamma+2\epsilon^{\prime}\gamma+\frac{d}{2}}.$

From (3.11),

$\displaystyle(2(1+\varepsilon)L+1)^{d}$
$\displaystyle=\sum_{j\in\mathbb{N}\setminus\\{0\\}\atop|n|\leq(1+\varepsilon)L}|\psi_{j,\omega}(n)|^{2}\geq\sum_{j:|n_{j,\omega}|\leq
L\atop|n|\leq(1+\varepsilon)L}|\psi_{j,\omega}(n)|^{2}$
$\displaystyle\geq\\#\\{j:|n_{j,\omega}|\leq L\\}\min_{j:|n_{j,\omega}|\leq
L}\sum_{|n|\leq(1+\varepsilon)L}|\psi_{j,\omega}(n)|^{2}$
$\displaystyle\geq\\#\\{j:|n_{j,\omega}|\leq
L\\}\left(1-\max_{j:|n_{j,\omega}|\leq
L}\sum_{|n|\geq(1+\varepsilon)L}|\psi_{j,\omega}(n)|^{2}\right)$
$\displaystyle\geq\\#\\{j:|n_{j,\omega}|\leq
L\\}(1-C(d,\epsilon^{\prime},\gamma)\varepsilon^{-\gamma+\frac{d}{2}}L^{-\gamma+2\epsilon^{\prime}\gamma+\frac{d}{2}}).$

Choose $\varepsilon=1/2$, $\epsilon^{\prime}=1/3$ and $\gamma=r/160$. Since
$r\geq 331d$, we have
$-\gamma+2\epsilon^{\prime}\gamma+\frac{d}{2}\leq-\frac{3}{16}d$. Then there
exists $L_{0}=L_{0}(d,r)$ large enough, such that

(3.12) $\\#\\{j:|n_{j,\omega}|\leq L\\}\leq C(d)L^{d},\quad\forall L\geq
L_{0},$

where $C(d)$ is independent of $L$ and $j$. This tells us that $L\geq L_{0}$,
$N(L)=\\#\\{j:|n_{j,\omega}|\leq L\\}$ is finite and we can reorder the
eigenfunctions so $|n_{j,\omega}|$ is increasing. Therefore, one has

$|n_{j,\omega}|\geq c(d)j^{\frac{1}{d}},$

for $j$ large enough (depending on $d$ and $r$). ∎

Finally, we can give the complete proof of Theorem 1.2.

######  _Proof of Theorem 1.2_.

Let $\phi\in\ell^{2}(\mathbb{Z}^{d})$ be such that, for some constant
$C_{\phi}>0$ and $\theta\geq r/160$, $|\phi(n)|\leq C_{\phi}|n|^{-\theta}$. We
have to bound $\|X^{q/2}e^{-\mathrm{i}\mathrm{H}_{\omega}t}\phi\|$, for some
$q>0$ and all $t>0$. Since
$e^{-\mathrm{i}\mathrm{H}_{\omega}t}\phi=\sum_{j}e^{-iE_{j,\omega}t}\langle\phi,\
\psi_{j,\omega}\rangle\psi_{j,\omega}$ and
$\|e^{-i\mathrm{H}_{\omega}t}\phi\|=\|\phi\|$, one has that

$\displaystyle\|X^{q/2}e^{-\mathrm{i}\mathrm{H}_{\omega}t}\phi\|^{2}$
$\displaystyle=\langle
X^{q}e^{-\mathrm{iH}_{\omega}t}\phi,e^{-\mathrm{iH}_{\omega}t}\phi\rangle$
$\displaystyle\leq\sum_{j}|\langle\phi,\psi_{j,\omega}\rangle|\ |\langle
X^{q}\psi_{j,\omega},\ e^{-\mathrm{iH}_{\omega}t}\phi\rangle|$
$\displaystyle\leq\|\phi\|\sum_{j}|\langle\phi,\
\psi_{j,\omega}\rangle|\|X^{q}\psi_{j,\omega}\|.$

From Theorem 3.1, we can choose $\epsilon^{\prime}=1/3$. Then there exists a
constant $C(\gamma)>0$ such that

(3.13) $|\psi_{j,\omega}(n)|\leq
C(\gamma)|n_{j,\omega}|^{\frac{\gamma}{3}}|n-n_{j,\omega}|^{-\gamma},\quad\quad\forall
n\in\mathbb{Z}^{d}.$

One has that

$\|X^{q}\psi_{j,\omega}\|^{2}=\sum_{n\in\mathbb{Z}^{d}}|n^{q}\psi_{j,\omega}(n)|^{2}\leq
C(\gamma)|n_{j,\omega}|^{\frac{2\gamma}{3}}\sum_{n\in\mathbb{Z}^{d}}|n|^{2q}|n-n_{j,\omega}|^{-2\gamma},$

where

$\displaystyle\sum_{n\in\mathbb{Z}^{d}}|n|^{2q}|n-n_{j,\omega}|^{-2\gamma}$
$\displaystyle\leq\sum_{|n-n_{j,\omega}|\geq
2|n_{j,\omega}|}|n|^{2q}|n-n_{j,\omega}|^{-2\gamma}+\sum_{|n-n_{j,\omega}|<2|n_{j,\omega}|}|n|^{2q}$
$\displaystyle\leq
C(d,q,\gamma)|n_{j,\omega}|^{-\gamma+q+\frac{d}{2}}+C(d,q)|n_{j,\omega}|^{2q+d}$
$\displaystyle\leq C(d,q,\gamma)|n_{j,\omega}|^{2q+d}.$

Therefore

$\|X^{q}\psi_{j,\omega}\|\leq
C(d,r,q,\gamma)|n_{j,\omega}|^{q+\frac{\gamma}{3}+\frac{d}{2}}.$

Moreover, from the assumption of $\phi$, we have

$|\langle\phi,\psi_{j,\omega}\rangle|\leq\sum_{n\in\mathbb{Z}^{d}}|\phi(n)||\psi_{j,\omega}(n)|\leq
C_{\phi}(\gamma)|n_{j,\omega}|^{\frac{\gamma}{3}}\sum_{n}|n|^{-\theta}|n-n_{j,\omega}|^{-\gamma}$

where

$\displaystyle\sum_{n}|n|^{-\theta}|n-n_{j,\omega}|^{-\gamma}$
$\displaystyle\leq\sum_{|n-n_{j,\omega}|<|n_{j,\omega}|/2}|n|^{-\theta}+\sum_{|n_{j,\omega}|\leq|n-n_{j,\omega}|\leq
2|n_{j,\omega}|}|n-n_{j,\omega}|^{-\gamma}$ $\displaystyle\ \ \
+\sum_{|n-n_{j,\omega}|\geq
2|n_{j,\omega}|}|n|^{-\theta}|n-n_{j,\omega}|^{-\gamma}$ $\displaystyle\leq
C(d,\theta)|n_{j,\omega}|^{-\theta+d}+C(d,\gamma)|n_{j,\omega}|^{-\gamma+d}+C(d,\theta,\gamma)|n_{j,\omega}|^{-\frac{\theta}{2}-\frac{\gamma}{2}+\frac{d}{2}}$
$\displaystyle\leq C(d,\theta,\gamma)|n_{j,\omega}|^{-\gamma+d}.$

Therefore

$|\langle\phi,\psi_{j,\omega}\rangle|\leq
C_{\phi}(d,\theta,\gamma)|n_{j,\omega}|^{-\frac{2\gamma}{3}+d}.$

Choose $0<q\leq\gamma/10$ and $\gamma=r/160$. Since $r\geq 1800d$, we have
$-\frac{\gamma}{3}+q+\frac{3d}{2}<-\frac{11d}{10}$. Recalling Lemma 3.3, one
has that

$\displaystyle\sum_{j}|\langle\phi,\
\psi_{j,\omega}\rangle|\|X^{q}\psi_{j,\omega}\|$ $\displaystyle\leq
C_{\phi}(d,r,q,\theta)\sum_{j}|n_{j,\omega}|^{-\frac{\gamma}{3}+q+\frac{3d}{2}}$
$\displaystyle\leq
C_{\phi}(d,r,q,\theta)\sum_{j}|n_{j,\omega}|^{-\frac{11d}{10}}$
$\displaystyle\leq C_{\phi}(d,r,q,\theta)\sum_{j}|j|^{-\frac{11}{10}}\leq
C_{\phi}(d,r,q,\theta).$

Therefore

$\|X^{q/2}e^{-\mathrm{iH}_{\omega}t}\phi\|^{2}\leq
C_{\phi}(d,r,q,\theta),\quad\forall t\geq 0.$

The proof of Theorem 1.2 is finished. ∎

## Acknowledgements

This work was supported by China Postdoctoral Science Foundation
(No.2021M692717).

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# Weyl-type bounds for twisted GL(2) short character sums

Aritra Ghosh  Aritra Ghosh
Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108,
India; email<EMAIL_ADDRESS>
###### Abstract.

Let f be a Hecke-Maass or holomorphic primitive cusp form for
$SL(2,\mathbb{Z})$ with normalized Fourier coefficients $\lambda_{f}(n)$. Let
$\chi$ be a primitive Dirichlet character of modulus p, where p is a prime
number. In this article we prove the following Weyl-type bound: for any
$\epsilon>0$,

$\sum_{|n|\ll
N}\lambda_{f}(n)\chi(n)\ll_{f,\epsilon}N^{3/4}p^{1/6}(pN)^{\epsilon}.$

##  1\. Introduction

Investigation of cancellation in sums of the forms

(1) $\mathfrak{S}_{\chi}(N)=\mathop{\sum}_{|n|\leq N}\chi(n),$

and

(2) $\mathfrak{S}_{f,\chi}(N)=\mathop{\sum}_{|n|\leq N}\lambda_{f}(n)\chi(n),$

has been a centre of attraction for mathematicians for a long time, where
$\chi$ is a character of conductor $p$ and $\lambda_{f}(n)$’s are normalized
Fourier coefficients of a Hecke-Maass or holomorphic primitive cusp form $f$
for $SL(2,\mathbb{Z})$. Suppose p is a prime.

In this context Polya-Vinogradov showed that
$\mathfrak{S}_{\chi}(N)\ll_{\epsilon}p^{\frac{1}{2}+\epsilon}$ which becomes
non-trivial if and only if $N>p^{\frac{1}{2}+\epsilon}$. As the smooth bump
functions are supported on a closed and bounded interval, so we can take
equation $(1)$ in a smooth way in the following sense. Let $W$ be a smooth
bump function supported on $[-2,2]$ and $W(x)=1$, for all $x\in[-1,1]$, so
upto a negligible error, equation $(1)$ and $(2)$ can be written as

(3)
$S_{\chi}(N)=\mathop{\sum}_{n\in\mathbb{Z}}\chi(n)W\left(\frac{n}{N}\right),$

and

(4)
$S_{f,\chi}(N)=\mathop{\sum}_{n\in\mathbb{Z}}\lambda_{f}(n)\chi(n)W\left(\frac{n}{N}\right),$

respectively.

Then equation $(3)$ becomes

(5) $S_{\chi}(N)=\frac{1}{2\pi
i}\mathop{\int}_{(\sigma)}N^{s}\widetilde{W}(s)L(s,\chi)ds$

If we shift the contour to $\sigma=1/2$, then roughly, from equation $(5)$,
one can say that

(6)
$S_{\chi}(N)\ll_{\epsilon}N^{\frac{1}{2}}|L(\frac{1}{2},\chi)|t^{\epsilon}.$

Here we have the convexity bound $L(1/2,\chi)\ll_{\epsilon}p^{1/4+\epsilon}$
for which $S_{\chi}(N)$ becomes non-trivial if and only if $N>p^{1/2}$, by
equation $(6)$. Hence subconvexity corresponds to cancellation in shorter
sums. Burgess (see, [1]) proved that
$L(1/2,\chi)\ll_{\epsilon}p^{3/16+\epsilon}$ which becomes non-trivial if and
only if $N>p^{3/4}$. Burgess’s method required new ideas, in particular it
uses the Riemann Hypothesis for curves over finite fields. Note that the
Burgess exponent of $3/16$ falls short of the exponent $1/6$ found by Weyl.
Curiously, the exponent $3/16$ often re-occurs in the modern incarnations of
these problems, see [2], [3], [4], [5] as examples.

For the $GL(2)$ case we have the following:

(7) $S_{f,\chi}(N)=\frac{1}{2\pi
i}\mathop{\int}_{(\sigma)}N^{s}\widetilde{W}(s)L(s,f\otimes\chi)ds.$

Again by shifting the contour, roughly one can say that,

(8)
$S_{f,\chi}(N)\ll_{f,\epsilon}N^{\frac{1}{2}}|L(\frac{1}{2},f\otimes\chi)|t^{\epsilon}.$

In this context, the convexity bound is
$L(\frac{1}{2},f\otimes\chi)\ll_{f,\epsilon}p^{\frac{1}{2}}$. So, using
equation $(8)$, we have, by convexity,
$S_{f,\chi}(N)\ll_{f,\epsilon}\sqrt{Np}$ which becomes non-trivial if and only
if $N>p$. Further improvement can be done. By Burgess’s exponent we have
$L(\frac{1}{2},f\otimes\chi)\ll_{f,\epsilon}p^{3/8}$ which gives
$S_{f,\chi}(N)\ll_{f,\epsilon}\sqrt{N}p^{\frac{3}{8}}<N\iff N>p^{3/4}$ which
becomes the Burgess’s range.

For the case of Dirichlet L-functions, the Burgess bound has only been
improved in some limited special cases. In a breakthrough, Conrey and
Iwaniec(see, [6]) obtained a Weyl quality bound for quadratic characters of
odd conductor using techniques from automorphic forms and Deligne’s solution
of the Weyl conjectures for varieties over finite fields. Another class of
results, such as [7] and [8], consider situations where the conductor $q$ of
$\chi$ runs over prime powers or otherwise has some special factorizations.
Notably, Milicevic (see, [9]) recently obtained a sub-Weyl subconvex bound
when $q=p^{n}$ with $n$ large. Also for the Weyl bound for short twisted sums
in the prime power case one can look the papers by V. Blomer and D. Milicevic
(see, [10]) and a related paper of R. Munshi and S. K. Singh (see, [11]).

In this paper we improve the range $N>p^{3/4}$ to the range $N>p^{2/3}$ and we
get a bound for $S_{f,\chi}(N)$ without going into the $L$-function. In this
paper we get the following result:

###  1.1. Statement of result

In this paper we prove the following Weyl-type bound:

######  Theorem 1.

Let f be a Hecke-Maass or holomorphic primitive cusp form for
$SL(2,\mathbb{Z})$. Let $\chi$ be a primitive Dirichlet character of modulus p
where p is a prime.Then for any $0<\theta<\frac{1}{10}$ with
$N^{1+2\theta}<p<N^{3/2-3\theta}$,

$S_{f,\chi}(N)\ll_{f,\epsilon}N^{3/4+\theta/2}p^{1/6}.$

From this we have the following corollary:

######  Corollary 1.

If $N>p^{\frac{2}{3(1-2\theta-4\delta)}}$ for some small $0<\delta<1$ then
$S_{f,\chi}(N)\ll_{f,\epsilon}N^{1-\delta+\epsilon}$ where $\epsilon$ is small
positive real number.

######  Application 1.

This result has a nice application. If the Fourier coefficients
$\lambda_{f}(n)$’s are real for all $n\in\mathbb{N}$ and $\chi$ is a quadratic
character then from theorem $1$, one can note that, $\lambda_{f}(n)$ and
$\chi(n)$ have opposite signs for atleast one $n\leq p^{2/3+\epsilon}$.

######  Remark 1.

In this paper ${}^{\prime}\ll^{\prime}$ means that whenever it occurs, the
implied constants will depend on $f,\epsilon$ only.

###  1.2. Sketch of proof

Here we shall describe our method briefly by taking $f$ to be a holomorphic
primitive cusp form for $SL(2,\mathbb{Z})$. Here we are using the method done
by Prof. R. Munshi (see, [12]). At first we consider the sum

$\mathfrak{S}_{f,\chi}(N)=\mathop{\sum}_{n\sim N}\lambda_{f}(n)\chi(n)$

for $N>p^{\frac{2}{3}+\delta}$ for some $\delta>0$, where $\lambda_{f}(n)$’s
are the normalized Fourier coeeficients of $f$ and $p$ is the conductor of
$\chi$. Here $p$ is taken to be a prime. Then we write this sum as

$\mathop{\sum\sum}_{n,m\sim N}\lambda_{f}(n)\chi(m)\delta_{n,m}$

where $\delta_{n,m}$ is the Kronecker $\delta$-symbol. Then we apply additive
harmonic - Jutila’s circle method (see section $(3)$), so that upto a
negligible error we get

$\mathfrak{S}_{f,\chi}(N)=\mathop{\sum\sum}_{n,m\sim
N}\lambda_{f}(n)\chi(m)\int_{\mathbb{R}}\tilde{I}(\alpha)e((n-m)\alpha)d\alpha.$

Here we need to save $N$. At first using $GL(2)$ Voronoi summation formula on
the $n$ sum we get that

$\sum_{n\sim
N}\lambda_{f}(n)e\left(\frac{na}{q}\right)e(n\alpha)\rightsquigarrow\sum_{n\sim\frac{Q^{2}}{N}}\lambda_{f}(n)e\left(\frac{-n\bar{a}}{q}\right),$

where $q$ is of size $Q=N^{1/2+\theta}$. Hence we save
$\sqrt{\frac{N}{N^{2\theta}}}$.

Now using Poisson summation formula on the $m$ sum we get

$\sum_{m\sim
N}\chi(m)e\left(-\frac{ma}{q}\right)\rightsquigarrow\frac{N}{pq}\sum_{|m|\ll\frac{pQ}{N}}\bar{\chi}(m)\chi(q)\mathbbm{1}_{a\equiv
m\bar{p}(\text{mod }q)},$

where $\mathbbm{1}$ is the indicator function for $a\equiv m\bar{p}(\text{mod
}q)$ on $\mathbb{Z}$. Here we save

$\frac{N}{\sqrt{pQ}}\times\sqrt{Q}=\frac{N}{\sqrt{p}}.$

So far we saved

$N^{1/2-\theta}\times\frac{N}{\sqrt{p}}=\frac{N^{3/2-\theta}}{\sqrt{p}}.$

So remains to save $\frac{\sqrt{p}}{N^{1/2-\theta}}$ in

$\sum_{q\in\Phi}\sum_{n\sim
N^{2\theta}}\sum_{m\sim\frac{pN^{\theta}}{\sqrt{N}}}\lambda_{f}(n)\bar{\chi}(m)\chi(q)e\left(-\frac{\bar{m}np}{q}\right).$

Now we choose $Q=Q_{1}Q_{2}$ and $q=q_{1}q_{2}$ in a certain way (see section
$(5)$). Then applying Cauchy-Schwarz inequality we have

$\sum_{q_{1}\sim
Q_{1}}\sum_{m\sim\frac{pn^{\theta}}{\sqrt{N}}}\Big{|}\sum_{n\sim
N^{2\theta}}\sum_{q_{2}\sim
Q_{2}}\lambda_{f}(n)\chi(q_{2})e\left(-\frac{\bar{m}np}{q_{1}q_{2}}\right)\Big{|}^{2}.$

Here for diagonal we save $Q_{2}N^{2\theta}$. Note that the diagonal sum
becomes okay if

$Q_{2}N^{2\theta}>\frac{p}{N^{1-2\theta}}\text{ i.e., }Q_{2}>\frac{p}{N}.$

Also for the off-diagonal we save

$\frac{pN^{\theta}}{\sqrt{N}\sqrt{Q_{1}}Q_{2}}=\frac{pN^{\theta/2}}{N^{3/4}\sqrt{Q_{2}}}.$

So this is satisfying if

$\frac{pN^{\theta/2}}{N^{3/4}\sqrt{Q_{2}}}>\frac{p}{N^{1-2\theta}}\iff
Q_{2}<N^{1/2-3\theta}.$

So we can choose $Q_{2}$ if

$\frac{p}{N}<N^{1/2-3\theta}\Rightarrow p<N^{3/2-3\theta}.$

Hence as long as $N>p^{2/3+\delta}$ for some $\delta>0$ and also we can get
$\theta$ and $Q_{2}$ and $N>p^{2/3}$ becomes the Weyl range.

##  2\. PRELIMINARIES

###  2.1. Preliminaries on holomorphic cusp forms.

For the sake of exposition we shall only present the case of holomorphic cusp
forms for $SL(2,\mathbb{Z})$. The case of Maass forms is just similar. Let
$f:\mathbb{H}\mapsto\mathbb{C}$, be a holomorphic cusp form with normalized
Fourier coefficients $\lambda_{f}(n)$. Also let us take $\chi$, a primitive
Dirichlet character of modulus $p$ where $p$ is a prime. Here one can note
that $|\lambda_{f}(n)|\ll_{f,\epsilon}N^{\epsilon}$ for $1\leq n\leq N$. Note
that, for holomorphic cusp forms, we have $\lambda_{f}(n)=0$ for all $n\leq
0$.

###  2.2. Voronoi summation formula.

We shall use the following Voronoi summation formula.

######  Lemma 2.1.

Let f be as above, and $v$ be a compactly supported smooth function on
$(0,\infty)$. Also consider $(a,q)=1$. Then we have

(9)
$\mathop{\sum}_{n=1}^{\infty}\lambda_{f}(n)e_{q}(an)v(n)=\frac{1}{q}\mathop{\sum}_{n=1}^{\infty}\lambda_{f}(n)e_{q}(-\bar{a}n)V(n),$

where $\bar{a}$ is the multiplicative inverse of $a\text{ mod q}$, and $V(n)$
is a certain integral Hankel transform of $v$.

Here note that, if we take $v$ to be supported in $[Y,2Y]$ and satisfying
$y^{j}v^{(j)}(y)\ll_{j}1$, then one can see that the sum on the right hand
side of $(9)$ becomes supported essentially on $n\ll{q^{2}(qY)^{\epsilon}/Y}$
(the implied constant depends on $f\text{ and }\epsilon$). Also note that the
terms with $n\gg{q^{2}(qY)^{\epsilon}/Y}$ contributes an amount which is
negligibly small. For smaller values of $n$ one can consider the trivial bound
$V(n/q^{2})\ll Y$.

###  2.3. Circle Method :

Here in this paper we shall use Jutila’s circle method (see, [13], [14]). Let
us consider $\Phi\subset[Q,2Q],\text{ where }Q\geq 1$.

Again let us take $\delta$ such that
$\frac{1}{Q^{2}}\leq\delta\leq\frac{1}{Q}$.

Now if we take $L=\mathop{\sum}_{q\in\Phi}\phi(q)$ then roughly we have
$L\asymp Q^{2}$.

Now we set $\widetilde{I}(\alpha)=\frac{1}{2\delta
L}\mathop{\sum}_{q\in\Phi}\sum_{d(\text{ mod q})}^{\star}I_{d/q}(\alpha)$
where $I_{d/q}$ is the indicator function of the interval
$[\frac{d}{q}-\delta,\frac{d}{q}+\delta]$.

Then this becomes an approximation of $I_{[0,1]}$ in the following sense:

######  Lemma 2.2.

We have

$\int_{\mathbb{R}}\left|I(\alpha)-\widetilde{I}(\alpha)\right|^{2}d\alpha\ll\frac{Q^{2+\epsilon}}{\delta
L^{2}},$

where $I$ is the indicator function of $[0,1]$.

This is a consequence of the Parseval theorem from Fourier analysis.

##  3\. Application of circle method :

Let us apply the circle method directly to the smooth sum

$S(N)=\mathop{\sum}_{n\in\mathbb{Z}}\lambda_{f}(n)\chi(n)h_{1}\left(\frac{n}{N}\right).$

At first recall that the function $h_{1}$ is smooth, supported in $[1,2]$ with
$h_{1}^{(j)}(x)\ll_{j}1$. Now we shall approximate the above sum $S(N)$ using
Jutila’s circle method by the following sum :

$\tilde{S}(N)=\frac{1}{L}\mathop{\sum}_{q\in\Phi}\hskip
2.84544pt\sideset{}{{}^{*}}{\sum}_{a(\text{ mod
}q)}\mathop{\sum\sum}_{n,m\in\mathbb{Z}}\lambda_{f}(n)\chi(m)e\left(\frac{a(n-m)}{q}\right)F(n,m),$

where $e_{q}(x)=e^{2\pi ix/q}$, and

$F(x,y)=h_{1}\left(\frac{x}{N}\right)h_{2}\left(\frac{y}{N}\right)\frac{1}{2\delta}\int_{-\delta}^{\delta}e(\alpha(n-m))d\alpha.$

Here $h_{2}$ is another smooth function having compact support in
$(0,\infty)$, with $h_{2}(x)=1$ for $x$ in the support of $h_{1}$. Also we
choose $\delta=N^{-1}$ so that we have

$\frac{\partial^{i+j}}{\partial^{i}x\partial^{j}y}F(x,y)\ll_{i,j}\frac{1}{N^{i+j}}.$

Then we have the following lemma :

######  Lemma 3.1.

Let $\Phi\subset[Q,2Q]$, with

$L=\sum_{q\in\Phi}\phi(q)\gg Q^{2-\epsilon},$

and $\delta\gg\frac{N^{2\theta}}{Q^{2}}$. Then we must have

$S(N)=\tilde{S}(N)+O_{f,\epsilon}\left(\sqrt{N}\frac{N(QN)^{\epsilon}}{Q}\right).$

###### Proof.

Let us define

$G(x):=\mathop{\sum\sum}_{n,m\in\mathbb{Z}}\lambda_{f}(n)\chi(m)h_{1}\left(\frac{n}{N}\right)h_{2}\left(\frac{m}{N}\right)e(x(n-m)).$

Here one can noth that

$S(N)=\int_{R}G(x)dx,$

and also

$\tilde{S}(N)=\int_{\mathbb{R}}\tilde{I}_{\Phi,\delta}(x)G(x)dx.$

Therefore we have

$\left|S(N)-\tilde{S}(N)\right|\leq\displaystyle\int_{\mathbb{R}}\left|I-\tilde{I}(x)\right|\left|\mathop{\sum}_{n\in\mathbb{Z}}\lambda_{f}(n)e(xn)h_{1}\left(\frac{n}{N}\right)\right|\left|\mathop{\sum}_{m\in\mathbb{Z}}\chi(m)e(-xm)h_{2}\left(\frac{m}{N}\right)\right|dx.$

For the sum in the middle if we use the point-wise bound

$\mathop{\sum}_{n\in\mathbb{Z}}\lambda_{f}(n)e(xn)h_{1}\left(\frac{n}{N}\right)\ll_{f,\epsilon}N^{\frac{1}{2}+\epsilon},$

then using Cauchy-Schwarz inequality we get that

$\left|S(N)-\tilde{S}(N)\right|\ll_{f,\epsilon}N^{\frac{1}{2}+\epsilon}\left[\displaystyle\int_{\mathbb{R}}\left|I-\tilde{I}(x)\right|^{2}dx\right]^{1/2}\left[\displaystyle\int_{\mathbb{R}}\left|\mathop{\sum}_{m\in\mathbb{Z}}\chi(m)e(-xm)h_{2}\left(\frac{m}{N}\right)\right|^{2}dx\right]^{1/2}.$

For the last sum, if we open the absolute value square and execute the
integral then we can note that we are left with the diagonal only, which has
size $N$. For the other sum if we use Lemma $2.2$, then it follows that

$\left|S(N)-\tilde{S}(N)\right|\ll_{f,\epsilon}\frac{(QN)^{1+\epsilon}}{\sqrt{\delta}L}\ll_{f,\epsilon}N\frac{\sqrt{N}(QN)^{\epsilon}}{Q}.$

∎

We shall choose the set of modulii in Section $5$. We shall pick the set of
the modulii to be $Q=N^{1/2+\theta}$. Hence the error term getting from the
previous lemma is $O(N^{1-\theta+\epsilon})$ for some $\theta>0$. Now we shall
proceed towards the estimation of $\tilde{S}(N)$.

##  4\. Estimation of $\tilde{S}(N)$

###  4.1. Applying Voronoi an Poisson summation

Let us now assume that each member of $\Phi$ is coprime to $p$, the modulus of
the character $\chi$. Let us define

(10) $\tilde{S}_{x}(N)=\frac{1}{L}\mathop{\sum}_{q\in\Phi}\hskip
5.69054pt\sideset{}{{}^{*}}{\sum}_{a(\text{mod
}q)}S\left(a,q,x,f\right)\,T\left(a,q,x,\chi\right),$

where

$S\left(a,q,x,f\right):=\sum_{n\in\mathbb{Z}}\lambda_{f}(n)h_{1}\left(\frac{n}{N}\right)e\left(\frac{an}{q}\right)e(nx),$

and

$T\left(a,q,x,\chi\right):=\sum_{m\in\mathbb{Z}}\chi(m)e\left(-\frac{am}{q}\right)h_{2}\left(\frac{m}{N}\right)e(-xm),$

with $|x|<\delta$. Then we have

$\tilde{S}(N)=(2\delta)^{-1}\int_{-\delta}^{\delta}\tilde{S}_{x}(N)dx.$

Now at first, let us consider the $n$-sum

(11)
$S\left(a,q,x,f\right)=\sum_{n=1}^{\infty}\lambda_{f}(n)h_{1}\left(\frac{n}{N}\right)e\left(\frac{an}{q}\right)e(nx).$

Then we have the following lemma:

######  Lemma 4.1.

Upto a negligible error, using Voronoi summation formula, on the $n$-sum we
have

(12)
$S\left(a,q,x,f\right)=\frac{N^{3/4}}{q^{1/2}}\sum_{|n|\ll\frac{Q^{2}}{N}}\frac{\lambda_{f}(n)}{n^{1/4}}e\left(-\frac{\bar{a}n}{q}\right)\mathcal{I}_{1}(n,x,q)+O(N^{-2021}),$

where $q\in[Q,2Q]$, coprime with $p$, where $\mathcal{I}_{1}(n,x,q)$ is given
by

$\mathcal{I}_{1}(n,x,q):=\int_{\mathbb{R}}h_{1}(y)e\left(Nxy\pm\frac{4\pi}{q}\sqrt{Nny}\right)W\left(\frac{4\pi\sqrt{Nny}}{q}\right)dy,$

where $W$ is a smooth nice function.

###### Proof.

If we apply Voronoi summation formula $(9)$ to the $n$-sum of the equation
$(11)$, then we have

$\sum_{n\in\mathbb{Z}}\lambda_{f}(n)e\left(\frac{an}{q}\right)e(nx)h_{1}\left(\frac{n}{N}\right)=\frac{1}{q}\sum_{n\in\mathbb{Z}}\lambda_{f}(n)e\left(-\frac{\bar{a}n}{q}\right)$
$\hskip
142.26378pt\times\int_{\mathbb{R}}h_{1}\left(\frac{y}{N}\right)e(xy)J_{k-1}\left(\frac{4\pi\sqrt{ny}}{q}\right)dy,$

where $J_{k-1}$ is the Bessel function. By changing $y\mapsto Ny$ and using
the approximation,

$J_{k-1}(x)\approx\frac{W(x)}{\sqrt{x}}e(x)+\frac{\bar{W}(x)}{\sqrt{x}}e(-x),$

where $W(x)$ is a nice function, the above equation becomes

$=\frac{N^{3/4}}{q^{1/2}}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{1/4}}e\left(-\frac{\bar{a}n}{q}\right)\mathcal{I}_{1}(n,x,q),$

where

$\mathcal{I}_{1}(n,x,q):=\int_{\mathbb{R}}h_{1}(y)e\left(Nxy\pm\frac{4\pi}{q}\sqrt{Nny}\right)W\left(\frac{4\pi\sqrt{Nny}}{q}\right)dy.$

Note that, this integral is negligibly small unless $|n|\ll\frac{Q^{2}}{N}$.

So we have

$\sum_{|n|\ll\frac{Q^{2}}{N}}\frac{\lambda_{f}(n)}{n^{1/4}}e\left(-\frac{\bar{a}n}{q}\right)\mathcal{I}_{1}(n,x,q)=\sum_{|n|\ll\frac{Q^{2}}{N}}\frac{\lambda_{f}(n)}{n^{1/4}}e\left(-\frac{\bar{a}n}{q}\right)\mathcal{I}_{1}(n,x,q)+O(N^{-2021}).$

This completes the proof of the lemma. ∎

######  Remark 2.

Note that $x\asymp\frac{\sqrt{n}}{\sqrt{N}Q}$, otherwise
$\mathcal{I}_{1}(n,x,q)$ is negligibly small.

Now let us consider the $m$-sum of $(10)$ given by

(13)
$T\left(a,q,x,\chi\right)=\sum_{m\in\mathbb{Z}}\chi(m)e\left(-\frac{am}{q}\right)h_{2}\left(\frac{m}{N}\right)e(-xm),$

then we get the following lemma:

######  Lemma 4.2.

Upto a negligible error, we get that

(14)
$T\left(a,q,x,\chi\right)=\frac{N\tau_{\chi}}{p}\sum_{\begin{subarray}{c}|m|\ll\frac{pQ}{N}\\\
m\bar{p}\equiv a(\text{mod
}q)\end{subarray}}\bar{\chi}(m)\chi(q)\mathcal{I}_{2}(m,x,q)+O(N^{-2021}),$

where

$\mathcal{I}_{2}(m,x,q):=\int_{\mathbb{R}}h_{2}(y)e(-Nxy)e\left(-\frac{mNy}{pq}\right)dy.$

###### Proof.

To the $m$-sum of the equation $(13)$, if we apply the Poisson summation
formula, then we get that

$T\left(a,q,x,\chi\right)=\frac{N}{pq}\sum_{m\in\mathbb{Z}}\mathcal{I}_{2}(m,x,q)\sum_{\beta\text{(
mod
}q)}\chi(\beta)e\left(-\frac{a\beta}{q}\right)e\left(\frac{m\beta}{pq}\right),$

where

$\mathcal{I}_{2}(m,x,q):=\int_{\mathbb{R}}h_{2}(y)e(-Nxy)e\left(-\frac{mNy}{pq}\right)dy.$

Here note that, this integral is negligibly small unless
$|m|\ll\frac{pQ}{N}N^{\epsilon}$.

So we have

$T\left(a,q,x,\chi\right)=\frac{N}{pq}\sum_{|m|\ll\frac{pQ}{N}N^{\epsilon}}\mathcal{I}_{2}(m,x,q)\sum_{\beta\text{(
mod
}q)}\chi(\beta)e\left(-\frac{a\beta}{q}\right)e\left(\frac{m\beta}{pq}\right)+O(N^{-2021}).$

As we know $(p,q)=1$, so that we can write $\beta$ as
$\beta=\beta_{1}q\bar{q}+\beta_{2}p\bar{p}$, where $\beta_{1},\beta_{2}$ are
the complete set of residue classes congruent to $p,q$ respectively. Then
substituting these in the place of $\beta$ we have

$T\left(a,q,x,\chi\right)=\frac{N\tau_{\chi}}{p}\sum_{\begin{subarray}{c}|m|\ll\frac{pQ}{N}\\\
m\bar{p}\equiv a(\text{mod
}q)\end{subarray}}\bar{\chi}(m)\chi(q)\mathcal{I}_{2}(m,x,q)+O(N^{-2021}).$

This completes the proof. ∎

From $(12)$ and $(14)$ we get that

######  Proposition 1.

(15)
$\begin{split}\tilde{S}_{x}(N)&=\frac{N^{7/4}}{\sqrt{p}L}\sum_{q\in\Phi}\frac{\chi(q)}{q^{1/2}}\sum_{|n|\ll\frac{Q^{2}}{N}}\sum_{\begin{subarray}{c}|m|\ll\frac{pQ}{N}\\\
(m,q)=1\end{subarray}}\frac{\lambda_{f}(n)}{n^{1/4}}\bar{\chi}(m)e\left(-\frac{p\bar{m}n}{q}\right)\mathcal{I}_{1}(n,x,q)\mathcal{I}_{2}(m,x,q)\\\
&\hskip 284.52756pt+O(N^{-2021}),\end{split}$

where $\mathcal{I}_{1}(n,x,q),\mathcal{I}_{2}(m,x,q)$ are given by $(12),(14)$
respectively.

##  5\. Further Estimation

At first let us consider $Q=Q_{1}Q_{2}$ and $q=q_{1}q_{2}$, where
$q_{1}\in[Q_{1},2Q_{1}]$ and $q_{2}\in[Q_{2},2Q_{2}]$, which are coprime with
each other and also with $p$. Also say
$N_{o}=\frac{Q^{2}}{N},M_{o}=\frac{pQ}{N}$. Here we note that, as
$0<\theta<1/10$ so that we have $Q_{2}>N_{o}$ and also we have $Q_{1}>N_{o}$.

Now if we apply Cauchy-Schwarz inequality on the equation $(15)$, then we have

(16)
$\begin{split}\tilde{S}_{x}(N)\ll\frac{N^{7/4}\sqrt{M_{o}}}{\sqrt{p}L\sqrt{Q_{1}}}\sum_{q_{1}\sim
Q_{1}}&\\\ &\times\left(\sum_{|m|\ll M_{o}}\left|\sum_{q_{2}\sim
Q_{2}}\frac{\chi(q_{2})}{q_{2}^{1/2}}\sum_{|n|\ll
N_{o}}\frac{\lambda_{f}(n)}{n^{1/4}}\mathcal{I}_{1}(n,x,q_{1}q_{2})\mathcal{I}_{2}(m,x,q_{1}q_{2})e\left(-\frac{p\bar{m}n}{q_{1}q_{2}}\right)\right|^{2}\right)^{1/2},\end{split}$

where $M_{o}=\frac{pQ}{N}$ and $N_{o}=\frac{Q^{2+\epsilon}}{N}$,

$\ll\frac{N^{7/4}\sqrt{M_{o}}}{\sqrt{p}L\sqrt{Q_{1}}}\sum_{q_{1}\sim
Q_{1}}\Omega(N_{o},q_{1},Q_{2},x)^{1/2},$

where $\Omega(N_{o},q_{1},Q_{2},x)$ is given by

$=\sum_{\begin{subarray}{c}|m|\ll M_{o}\\\
(m,q)=1\end{subarray}}\left|\sum_{q_{2}\sim
Q_{2}}\frac{\chi(q_{2})}{q_{2}^{1/2}}\sum_{|n|\ll
N_{o}}\frac{\lambda_{f}(n)}{n^{1/4}}\mathcal{I}_{1}(n,x,q_{1},q_{2})\mathcal{I}_{2}(m,x,q_{1},q_{2})e\left(-\frac{p\bar{m}n}{q_{1}q_{2}}\right)\right|^{2}.$

Now openning the brackets of $\Omega(N_{o},q_{1},Q_{2},x)$, we have

$\sum_{\begin{subarray}{c}|m|\in\mathbb{Z}\\\
(m,q)=1\end{subarray}}W^{\prime}\left(\frac{m}{M_{o}}\right)\sum_{q_{2},q_{2}^{\prime}\sim
Q_{2}}\frac{\chi(q_{2}\bar{q_{2}^{\prime}})}{(q_{2}q_{2}^{\prime})^{1/2}}\sum_{|n|,|n|^{\prime}\ll
N_{o}}\frac{\lambda_{f}(n)\lambda_{f}(n^{\prime})}{(nn^{\prime})^{1/4}}e\left(\frac{\bar{m}p(q_{2}^{\prime}n-n^{\prime}q_{2})}{q_{1}q_{2}q_{2}^{\prime}}\right)$
$\times\mathcal{I}_{1}(n,x,q_{1}q_{2})\mathcal{I}_{2}(m,x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n,x,q_{1}q_{2}^{\prime})\bar{\mathcal{I}_{2}}(m,x,q_{1}q_{2}^{\prime}),$

where $W^{\prime}$ is a smooth nice function with supp$(W^{\prime})$ is in
$[-2,2]$ such that $W(x)=1$ for $x\in[-1,1]$.

Now if we use Poisson summation formula to the $m$-sum then the $m$-sum
becomes

$\frac{M_{o}}{q_{1}q_{2}q_{2}^{\prime}}\sum_{m\in\mathbb{Z}}S(p(q_{2}^{\prime}n-n^{\prime}q_{2}),m;q_{1}q_{2}q_{2}^{\prime})\mathcal{I}(m,x,q_{1},q_{2},q_{2}^{\prime}),$

where

$\mathcal{I}(m,x,q_{1},q_{2},q_{2}^{\prime}):=\int_{\mathbb{R}}W^{\prime}(y)\mathcal{I}_{2}(M_{o}y,x,q_{1}q_{2})\bar{\mathcal{I}_{2}}(M_{o}y,x,q_{1}q_{2}^{\prime})e\left(-\frac{mM_{o}y}{q_{1}q_{2}q_{2}^{\prime}}\right).$

Here note that the integral $\mathcal{I}$ is negligilbly small unless
$|m|\ll\frac{Q_{1}Q_{2}^{2}}{M_{o}}=\frac{Q_{2}Q}{M_{o}}$.

Let $R_{o}=\frac{Q_{2}Q}{M_{o}}=\frac{NQ_{2}}{p}$.

So we get

(17)
$\tilde{S}_{x}(N)=\frac{N^{7/4}\sqrt{M_{o}}}{\sqrt{p}L\sqrt{Q_{1}}}\sum_{q_{1}\sim
Q_{1}}\Omega(N_{o},q_{1},Q_{2},x)^{1/2}+O(N^{-2021}).$

where $\Omega(N_{o},q_{1},Q_{2},x)$ is given by

$\Omega(N_{o},q_{1},Q_{2},x)=\frac{M_{o}}{q_{1}}\sum_{q_{2},q_{2}^{\prime}\sim
Q_{2}}\frac{\chi(q_{2}\bar{q_{2}^{\prime}})}{(q_{2}q_{2}^{\prime})^{3/2}}\sum_{|n|,|n|^{\prime}\ll
N_{o}}\frac{\lambda_{f}(n)\lambda_{f}(n^{\prime})}{(nn^{\prime})^{1/4}}\mathcal{I}_{1}(n,x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n,x,q_{1}q_{2}^{\prime})\\\
\times\mathcal{I}_{1}(n^{\prime},x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n^{\prime},x,q_{1}q_{2}^{\prime})\sum_{|m|\ll
R_{o}}S(p(q_{2}^{\prime}n-n^{\prime}q_{2}),m;q_{1}q_{2}q_{2}^{\prime})\mathcal{I}(m,x,q_{1},q_{2},q_{2}^{\prime}).$

######  Remark 3.

For the integrals as there are no analytic terms so we can consider trivial
bound for them, i.e., they are bounded by $1$. Also note that
$|\lambda_{f}(n)|\ll N^{\epsilon}$ for $|n|\leq N$ and $\sum_{|n|\ll
N_{o}}\frac{1}{n^{1/4}}\ll N_{o}^{3/4}$.

######  Lemma 5.1.

We have

(18) $\Omega(N_{o},q_{1},Q_{2},x)\ll
M_{o}N_{o}^{1/2}+N_{o}^{3/2}Q_{2}^{2}\sqrt{Q_{1}}.$

The proof of this lemma is given below. The first term of right hand side of
$(18)$ is coming from $m=0$ and the second term is coming from other $m$’s,
i.e., for the terms with $m\neq 0$. For the proof, at first we shall consider
the zero frequency case, i.e., when $m=0$.

### The zero frequency

The zero frequency $m=0$ has to be treated differntly. Let $\Sigma_{o}$ denote
the contribution of the zero frequency to $\tilde{S}_{x}(N)$, i.e.,

(19)
$\begin{split}\Sigma_{o}=\frac{M_{o}}{q_{1}}\sum_{q_{2},q_{2}^{\prime}\sim
Q_{2}}\frac{\chi(q_{2}\bar{q_{2}^{\prime}})}{(q_{2}q_{2}^{\prime})^{3/2}}\sum_{|n|,|n|^{\prime}\ll
N_{o}}\frac{\lambda_{f}(n)\lambda_{f}(n^{\prime})}{(nn^{\prime})^{1/4}}S(p(q_{2}^{\prime}n-n^{\prime}q_{2}),0;q_{1}q_{2}q_{2}^{\prime})\\\
&\times\mathcal{I}_{1}(n^{\prime},x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n^{\prime},x,q_{1}q_{2}^{\prime})\mathcal{I}_{1}(n,x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n,x,q_{1}q_{2}^{\prime})\mathcal{I}(m,x,q_{1},q_{2},q_{2}^{\prime})\\\
&=\frac{M_{o}}{q_{1}}\sum_{q_{2},q_{2}^{\prime}\sim
Q_{2}}\frac{\chi(q_{2}\bar{q_{2}^{\prime}})}{(q_{2}q_{2}^{\prime})^{3/2}}\sum_{|n|,|n|^{\prime}\ll
N_{o}}\frac{\lambda_{f}(n)\lambda_{f}(n^{\prime})}{(nn^{\prime})^{1/4}}\sum_{d|q_{1}q_{2}q_{2}^{\prime}}d\mu\left(\frac{p(q_{2}^{\prime}n-n^{\prime}q_{2})}{d}\right)\\\
&\times\mathcal{I}_{1}(n^{\prime},x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n^{\prime},x,q_{1}q_{2}^{\prime})\mathcal{I}_{1}(n,x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n,x,q_{1}q_{2}^{\prime})\mathcal{I}(m,x,q_{1},q_{2},q_{2}^{\prime})\\\
&=\frac{M_{o}}{q_{1}}\sum_{q_{2},q_{2}^{\prime}\sim
Q_{2}}\frac{\chi(q_{2}\bar{q_{2}^{\prime}})}{(q_{2}q_{2}^{\prime})^{3/2}}\sum_{d|q_{1}q_{2}q_{2}^{\prime}}d\sum_{\begin{subarray}{c}|n|,|n^{\prime}|\ll
N_{o}\\\ q_{2}^{\prime}n-n^{\prime}q_{2}\equiv 0(\text{mod
d})\end{subarray}}\frac{\lambda_{f}(n)\lambda_{f}(n^{\prime})}{(nn^{\prime})^{1/4}}\\\
&\times\mathcal{I}_{1}(n,x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n,x,q_{1}q_{2}^{\prime})\mathcal{I}(m,x,q_{1},q_{2},q_{2}^{\prime})\mathcal{I}_{1}(n^{\prime},x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n^{\prime},x,q_{1}q_{2}^{\prime}).\end{split}$

######  Lemma 5.2.

We have

(20) $\Sigma_{o}\ll M_{o}N_{o}^{1/2}.$

###### Proof.

In the case for $m=0$ we have six cases according to the divisors of
$q_{1}q_{2}q_{2}^{\prime}$ and note that $Q=Q_{1}Q_{2}$.

#### Case 1

Let $d=q_{1}q_{2}q_{2}^{\prime}$. Then note that size of $d$ for this case is
$Q_{1}Q_{2}^{2}$. But size of $q_{2}^{\prime}n-n^{\prime}q_{2}$ is
$Q_{2}N_{o}$. So for this case

$q_{2}^{\prime}n-n^{\prime}q_{2}\equiv 0(\text{mod d})\iff
q_{2}=q_{2}^{\prime}\text{ and }n=n^{\prime},$

as size of $n$ is smaller than size of $Q_{2}$. Hence we have

$\Sigma_{o}\ll M_{o}N_{o}^{1/2}.$

#### Case 2

Let $d=1$. Then we get that

$\Sigma_{o}\ll\frac{M_{o}N_{o}^{3/2}}{Q}.$

But as $N_{o}\ll Q$ so for this case, we must have,

$\Sigma_{o}\ll M_{o}N_{o}^{1/2}.$

#### Case 3

Let $d=q_{1}$. For this case, we have

$\Sigma_{o}\ll\frac{M_{o}N_{o}^{3/2}}{Q_{2}}.$

But as $N_{o}<Q_{2}$ so that for this case again we have,

$\Sigma_{o}\ll M_{o}N_{o}^{1/2}.$

#### Case 4

Now consider $d=q_{2}$. For this case we have

$\Sigma_{o}\ll\frac{M_{o}N_{o}^{3/2}}{Q_{1}}.$

But as $N_{o}<Q_{1}$ so for this case, we must have,

$\Sigma_{o}\ll M_{o}N_{o}^{1/2}.$

For the case $d=q_{2}^{\prime}$ we have to process similarly and we shall get
the same bound.

#### Case 5

Now take $d=q_{1}q_{2}$. But as size of $q_{2}^{\prime}n-n^{\prime}q_{2}$ is
$Q_{2}N_{o}$ which is less than the size of $d$, i.e., $Q_{1}Q_{2}$ so for
this case we have

$q_{2}^{\prime}n-n^{\prime}q_{2}\equiv 0(\text{mod d})\iff
q_{2}=q_{2}^{\prime}\text{ and }n=n^{\prime}.$

Hence we have

$\Sigma_{o}\ll\frac{M_{o}N_{o}^{1/2}}{Q_{2}}.$

But then again we have, for this case,

$\Sigma_{o}\ll M_{o}N_{o}^{1/2}.$

For $d=q_{1}q_{2}^{\prime}$ if we process similarly then we shall get the same
bound.

#### Case 6

For the last case we have $d=q_{2}^{2}$. This case will be similar as case
$(5)$. By considering the size of $d$ for this case again we can say that

$q_{2}^{\prime}n-n^{\prime}q_{2}\equiv 0(\text{mod d})\iff
q_{2}=q_{2}^{\prime}\text{ and }n=n^{\prime}.$

So for this case we have

$\Sigma_{o}\ll\frac{M_{o}N_{o}^{1/2}}{Q_{1}}.$

So again we have, for this case,

$\Sigma_{o}\ll M_{o}N_{o}^{1/2}.$

This completes the proof of the lemma $5.2$. ∎

### Non-zero frequency

Now we shall consider the non-zero frequency case, i.e., when $m\neq 0$. For
$m\neq 0$ we use Weil bound for Kloosterman sum so that we get

(21) $\begin{split}\Sigma_{\neq
0}=\frac{M_{o}}{q_{1}}\sum_{q_{2},q_{2}^{\prime}\sim
Q_{2}}\frac{\chi(q_{2}\bar{q_{2}^{\prime}})}{(q_{2}q_{2}^{\prime})^{3/2}}\sum_{|n|,|n|^{\prime}\ll
N_{o}}\frac{\lambda_{f}(n)\lambda_{f}(n^{\prime})}{(nn^{\prime})^{1/4}}\\\
&\times\mathcal{I}_{1}(n,x,q_{1}q_{2})\bar{\mathcal{I}_{1}}(n,x,q_{1}q_{2}^{\prime})\mathcal{I}(m,x,q_{1},q_{2},q_{2}^{\prime})\sum_{1\leq
m\ll
R_{o}}S(p(q_{2}^{\prime}n-n^{\prime}q_{2}),m;q_{1}q_{2}q_{2}^{\prime}).\end{split}$

Now by the Weil’s bound for Kloosterman sums we have

$\sum_{1\leq|m|\ll
R_{o}}S(p(q_{2}^{\prime}n-n^{\prime}q_{2}),m;q_{1}q_{2}q_{2}^{\prime})\ll
Q_{2}\sqrt{Q_{1}}\sum_{1\leq|m|\ll
R_{o}}(p(q_{2}^{\prime}n-n^{\prime}q_{2}),m;q_{1}q_{2}q_{2}^{\prime})^{1/2}.$

######  Lemma 5.3.

For any $x,y\in\mathbb{R}\text{ with }x,y\geq 1$ and $c\in\mathbb{N}$, we
have,

$\sum_{1\leq a\leq x}\sum_{1\leq b\leq y}(a,b,c)\leq xy.$

###### Proof.

Let $(a,b)=d$ so that $d\geq 1$. Then we have

$\sum_{1\leq a\leq x}\sum_{1\leq b\leq y}(a,b,c)\leq\sum_{1\leq a\leq
x}\sum_{1\leq b\leq y}(a,b)\leq\sum_{1\leq d\leq x}d\sum_{1\leq
b\leq\frac{y}{d}}1=xy.$

This completes the proof of this lemma. ∎

By the previous lemma, we have,

$\sum_{1\leq|m|\ll
R_{o}}(p(q_{2}^{\prime}n-n^{\prime}q_{2}),m;q_{1}q_{2}q_{2}^{\prime})^{1/2}\ll
R_{o}.$

Hence we get that

(22) $\sum_{|m|\ll
R_{o}}S(p(q_{2}^{\prime}n-n^{\prime}q_{2}),m;q_{1}q_{2}q_{2}^{\prime})\mathcal{I}(m,x,q_{1},q_{2},q_{2}^{\prime})\ll
R_{o}Q_{2}\sqrt{Q_{1}}.$

Now putting values of $R_{o}\text{ and }M_{o}$ we get that,

$\Sigma_{\neq
0}\ll\frac{pQ}{NQ_{1}}\times\frac{NQ_{2}^{2}\sqrt{Q_{1}}}{p}\times\frac{1}{Q_{2}}\times
N_{o}^{3/2},$ $\text{ i.e., }\Sigma_{\neq 0}\ll
N_{o}^{3/2}Q_{2}^{2}\sqrt{Q_{1}}.$

This completes the proof of the lemma $5.1$.

##  6\. Final estimation

From the equations $(17)\text{ and }(18)$ we get that

(23)
$\begin{split}\tilde{S}_{x}(N)&\ll\frac{N^{7/4}\sqrt{M_{o}}}{\sqrt{p}L\sqrt{Q_{1}}}\sum_{q_{1}\sim
Q_{1}}\left(M_{o}N_{o}^{1/2}+N_{o}^{3/2}Q_{2}^{2}\sqrt{Q_{1}}\right)^{1/2}\\\
&\ll\frac{N^{7/4}\sqrt{Q_{1}M_{o}}}{\sqrt{p}L}\left(M_{o}^{1/2}N_{o}^{1/4}+N_{o}^{3/4}Q_{2}Q_{1}^{1/4}\right).\end{split}$

Now for any given $\theta$, the optimum choice of $Q_{1}$ is obtained by
equating the two terms of equation $(18)$ and using the relation
$Q_{1}Q_{2}=Q=N^{1/2+\theta}$, so that we have

(24) $Q_{1}=\frac{N^{1+2\theta}}{p^{2/3}}.$

This satisfies our requirement that $\frac{p}{N}<Q_{2}<N^{1/2-3\theta}$. Now
putting this value of $Q_{1}$ in the equation $(23)$, we get that

$\tilde{S}_{x}(N)\ll N^{3/4+\theta/2}p^{1/6}.$

This completes the proof of the theorem.

##  7\. Acknowledgements

Author is grateful to Prof. Ritabrata Munshi for sharing his ideas with him
and for his support and encouragement. Author is also thankful to Mallesham,
S. Kumar, S. K. Singh and P. Sharma for their constant support and
encouragements. Finally, author would like to thank Stat-Math unit, Indian
Statistical Institute, Kolkata, for providing excellent research environment.

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