# Bounds for Smooth Theta Sums
with Rational Parameters

Francesco Cellarosi111Department of Mathematics and Statistics. Queen’s
University. Kingston, ON, Canada. and Tariq Osman222Department of Mathematics.
Brandeis University. Waltham, MA, U.S.A.

###### Abstract

We provide an explicit family of pairs
$({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})\in\mathbb{R}^{k}\times\mathbb{R}^{k}$
such that for sufficiently regular $f$, there is a constant $C$ for which the
theta sum bound

$\displaystyle\left|\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}f\\!\left(\tfrac{1}{N}{\text{\boldmath$n$}}\right)\exp\\{2\pi
i((\tfrac{1}{2}\|{\text{\boldmath$n$}}\|^{2}+{\text{\boldmath$\beta$}}\cdot{\text{\boldmath$n$}})x+{\text{\boldmath$\alpha$}}\cdot{\text{\boldmath$n$}})\\}\right|\leq
CN^{k/2},$

holds for every $x\in\mathbb{R}$ and every $N\in\mathbb{N}$. Central to the
proof is realising that, for fixed $N$, the theta sum normalised by $N^{k/2}$
agrees with an automorphic function $|\Theta_{f}|$ evaluated along a special
curve known as a horocycle lift. The lift depends on the pair
$({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})$, and so the bound
follows from showing that there are pairs such that $|\Theta_{f}|$ remains
bounded along the entire horocycle lift.

###### Contents

  1. 1 Introduction
  2. 2 Representation-Theoretical Preliminaries
    1. 2.1 The Heisenberg Group its Schrödinger Representation
    2. 2.2 $\mathrm{Sp}(k,\mathbb{R})$ and its Projective Shale-Weil Representation
    3. 2.3 The Jacobi Group and its Projective Schrödinger-Weil Representation
  3. 3 The Jacobi Theta Function
    1. 3.1 Writing $|\Theta_{f}|$ in coordinates on $\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2k}$
    2. 3.2 Regular Cut-off Functions
    3. 3.3 Invariance Properties
    4. 3.4 Fundamental Domains
  4. 4 Uniform Bounds
    1. 4.1 Uniform Bounds for fixed $\xi$
    2. 4.2 Bounds over $\Gamma$-orbits
    3. 4.3 Study of $\Gamma$-Orbits and Conclusions
  5. 5 Applications
    1. 5.1 Uniform Bounds for $\vartheta_{3}$
    2. 5.2 Estimates for $S_{N}$ independent of $x$

##  1 Introduction

Let $\mathrm{e}\\!\left(z\right):=e^{2\pi iz}$. Let
$f:\mathbb{R}^{k}\to\mathbb{R}$ be a Schwartz function. We consider the
_generalised quadratic Weyl sum_ (or _generalised theta sums_)

$\displaystyle
S^{f}_{N}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}}):=\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}f\\!\left(\tfrac{1}{N}{\text{\boldmath$n$}}\right)\mathrm{e}\\!\left((\tfrac{1}{2}\|{\text{\boldmath$n$}}\|^{2}+{\text{\boldmath$\beta$}}\cdot{\text{\boldmath$n$}})x+{\text{\boldmath$\alpha$}}\cdot{\text{\boldmath$n$}}\right),$
(1.1)

where $x\in\mathbb{R}$, $N\in\mathbb{N}$, and
${\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}}\in\mathbb{R}^{k}$. We
can think of $f$ as a smooth cut-off function. We prove the following theorem.

######  Theorem 1.1 (Main Theorem, for Schwartz cut-offs.).

Let $f\in\mathcal{S}(\mathbb{R}^{k})$ be a Schwartz function and let
${\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}}\in\mathbb{R}^{k}$. If
${\text{\boldmath$\alpha$}}=(\alpha_{1},\cdots,\alpha_{n})$,
${\text{\boldmath$\beta$}}=(\beta_{1},\cdots,\beta_{n})$, and at least one of
$(\alpha_{i},\beta_{i})=(\tfrac{a}{2m},\tfrac{b}{2m})\in\mathbb{Q}^{2}$ is
such that $\gcd(a,b,m)=1$, and $a$, $b$, and $m$ are all odd, then

$\displaystyle\left|S_{N}^{f}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})\right|\ll_{k,m,{\text{\boldmath$\beta$}},f}N^{k/2},$
(1.2)

for every $x\in\mathbb{R}$ and every $N\in\mathbb{N}$.

In (1.2) and in the rest of the paper we use Vinogradov’s “$\ll$” notation
(which is equivalent to Landau’s $O$-notation) and stress the dependence of
the implied constants upon the parameters written as subscripts. As we shall
see, we can relax the assumption that $f$ is Schwartz, see Theorem 4.9. When
$k=1$ we have the following corollary.

######  Corollary 1.2.

Let $f\in\mathcal{S}(\mathbb{R})$. Let $\alpha=\tfrac{a}{2m}$ and
$\beta=\tfrac{b}{2m}$, with $a$, $b$, and $m$ all odd, and such that
$\gcd(a,b,m)=1$. Then

$\displaystyle|S_{N}^{f}(x;\alpha,\beta)|\ll_{m,\beta,f}\sqrt{N},$ (1.3)

for every $x\in\mathbb{R}$ and every $N\in\mathbb{N}$.

######  Remark 1.3.

The only previously known instance of Corollary 1.2 is due to Marklof when
$m=1$, i.e. $(\alpha,\beta)=(\frac{1}{2},\frac{1}{2})$, see Section 5.2 of
[18]. Furthermore, it follows from Theorem 1.4 (i) in [4] and Theorem 1.0.7 in
[5] that the limiting distribution of
$|N^{-\tfrac{1}{2}}S_{N}(x;\alpha,\beta)|$ as $N\to\infty$ for any pair
$(\alpha,\beta)\in\mathbb{R}^{2}$ _not_ of the form given in Corollary 1.2, is
heavy tailed. Therefore, the rational pairs $(\alpha,\beta)$ in Corollary 1.2
are _the only pairs_ in $\mathbb{R}^{2}$ for which the bound (1.3) holds for
every $x\in\mathbb{R}$ and every $N\in\mathbb{N}$.

In Section 5.1, as an illustration of the main theorem, we also obtain bounds
for the classical Jacobi theta function

$\displaystyle\vartheta_{3}(z;w)=\sum_{n\in\mathbb{Z}}e^{2\pi inz+\pi
n^{2}w},$ (1.4)

where $z\in\mathbb{C}$ and $w\in\mathfrak{H}:=\\{x+iy\in\mathbb{C}:y>0\\}$.
When $w$ approaches the boundary in $\mathfrak{H}$ and $z$ is of the form
$\alpha+\beta\operatorname{Re}w$ with $(\alpha,\beta)$ as in Corollary 1.2, we
have the following

######  Theorem 1.4.

Suppose
$(\alpha,\beta)=(\tfrac{a}{2m},\tfrac{b}{2m})\in\mathbb{R}^{2}/\mathbb{Z}^{2}$
with $\gcd(a,b,m)=1$ and $a,b$ and $m$ all odd. Then

$\displaystyle\left|\vartheta_{3}(\alpha+\beta
x,x+i\varepsilon)\right|\ll_{m}\varepsilon^{-1/4}.$ (1.5)

Figure 1 illustrates Theorem 1.4 for
$(\alpha,\beta)=(\frac{1}{2},\frac{1}{6})$ (that is $a=3,b=1,m=3$) for various
values of $x$ and of $\varepsilon=N^{-2}$. For contrast, Figure 2 shows the
lack of a uniform bound for
$\varepsilon^{1/4}\vartheta_{3}(\frac{1}{2}+\frac{x}{3},x+i\varepsilon)$ as
$\varepsilon=N^{-2}$ decreases. In this case the pair
$(\alpha,\beta)=(\frac{1}{2},\frac{1}{3})$ does not satisfy the hypotheses of
Theorem 1.4 since $a=3,b=2,m=3$ are not all odd.

Figure 1: Illustration of Theorem 1.4. Curves
$x\mapsto\varepsilon^{1/4}\vartheta_{3}(\frac{1}{2}+\frac{x}{6},x+i\varepsilon)\in\mathbb{C}$
with $\varepsilon=N^{-2}$ and $N=100$ (top row) or $N=1000$ (bottom row). Each
panel indicates the range of $x$. Figure 2: Curves $[0.2,0.23]\ni
x\mapsto\varepsilon^{1/4}\vartheta_{3}(\frac{1}{2}+\frac{x}{3},x+i\varepsilon)\in\mathbb{C}$
with $\varepsilon=N^{-2}$ and $N\in\\{100,200,300,400\\}$. The pair
$(\frac{1}{2},\frac{1}{3})$ does not satisfy the hypotheses of Theorem 1.4.

There are many results in the literature concerning upper bounds for
generalised theta sums as defined in (1.1). These estimates have modern
applications, for instance, in [2], [10], [16], [17] they are used to
understand the value distribution of quadratic forms.

Estimates for $S_{N}:=S_{N}^{f}$ where $f=\mathds{1}_{(0,1]},k=1$ have
received much attention over the years. In the case where
$x,\alpha,\beta\in\mathbb{Q}$, $S_{N}$ reduces to a quadratic Gauss sum for
which various bounds are classical, e.g. if $\gcd(a,q)=1$ and $N\leq q$ then
$|S_{N}(\tfrac{a}{q};0,0)|=O(\sqrt{q})$. See [12], [21] for further details.

The detailed study of $S_{N}$ for $x\in\mathbb{R}$ was initiated by Hardy and
Littlewood in [11], who were attracted by its “interesting and beautiful
properties”. In particular, they prove an approximate functional equation
which they use to obtain various bounds for $S_{N}$, typically with some
restriction on $x$, e.g. for $x$ of bounded type they prove that
$S_{N}(x;\alpha,0)\ll_{x}\sqrt{N}$.

For generic $x$ it is known that $|S_{N}(x;\alpha,\beta)|\ll_{x}\sqrt{N}\log
N$, first in the case where $(\alpha,\beta)=(0,0)$ by Fiedler, Jurkat and
Körner [7], and then for every $(\alpha,\beta)\in\mathbb{R}^{2}$ by Flaminio
and Forni [8], even with a reduction of the power of $\log N$, see Remark 5.4.
Analogous bounds for theta sums in higher rank, including the were obtained by
Cosentino and Flaminio [6], with recent improvements by Marklof and Welsh
[19], [20]. For bounds on Weyl sums of arbitrary degree (not just theta sums,
in which the degree of the polynomial in the exponential sum is $2$), see the
recent work of Flaminio and Forni [9] and the references therein.

In light of these results, we see that the behaviour of
$|S^{f}_{N}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})|$ for
$({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})$ as in Theorem 1.1 and
Corollary 1.2 is far from typical. In particular, the implied constant in
(1.3) is independent of $x$. Therefore, for $(\alpha,\beta)$ as in the
statement of Corollary 1.2 there exists $R_{0}>0$ such that for any $R>R_{0}$

$\displaystyle\mathrm{Leb}\\{x\in\mathbb{R}:\tfrac{1}{\sqrt{N}}S^{f}_{N}(x;\alpha,\beta)>R\\}=0,$
(1.6)

for every $N$. It follows that for $R>R_{0}$

$\displaystyle\lim_{N\to\infty}\mathrm{Leb}\\{x\in\mathbb{R}:\tfrac{1}{\sqrt{N}}S^{f}_{N}(x;\alpha,\beta)>R\\}=0,$
(1.7)

and so the limiting distribution (as $N\to\infty$) of
$\tfrac{1}{\sqrt{N}}S^{f}_{N}(x;\alpha,\beta)$ for random $x$ must be
compactly supported in this case. In this way we see that Theorem 1.1 implies
Theorem 1.0.7 (i) in [5].

Our general approach aligns with that of [14], [15], [17] which interprets
$\tfrac{1}{\sqrt{N}}|S^{f}_{N}|$ as an automorphic function $|\Theta_{f}|$
evaluated along special curves, known as horocycle lifts.

In Section 2 we outline the construction of the (projective) Shrödinger-Weil
representation, a unitary representation of Jacobi group on
$\mathrm{L}^{2}(\mathbb{R}^{k})$. We then use this representation in Section 3
to define a special real-valued function $|\Theta_{f}|$ over the group
$G:=\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2k}$, which is a subgroup of
the Jacobi group, and $f:\mathbb{R}\to\mathbb{R}$ is some sufficiently regular
weight function. The sum $\tfrac{1}{\sqrt{N}}|S^{f}_{N}|$ agrees with
$|\Theta_{f}|$ along special curves
$\mathcal{C}^{({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})}_{N}$,
known as horocycle lifts. Special invariance properties of $|\Theta_{f}|$
imply that it can be viewed as a function on a non-compact, finite volume
homogeneous space $\Gamma\backslash G$. In Section 4 we prove a uniform bound
for $|\Theta_{f}|$, provided certain parameters avoid certain (explicit)
regions of growth in $\Gamma\backslash G$. We then give a simple, explicit
condition on $({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})$ so that
the curve
$\Gamma\mathcal{C}^{({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})}_{N}$
(a horocycle lift viewed as a curve in $\Gamma\backslash G$) is bounded away
from the regions of growth, uniformly in $N$. In Section 4.3 we give explicit
conditions on the pair
$({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})$ so that
$\Gamma\mathcal{C}^{({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})}_{N}$
avoids regions of growth, yielding Theorem 1.1.

The final Section 5 contains applications of our main theorem to produce a
bound for the classical Jacobi theta function $\vartheta_{3}$ (Theorem 1.4)
and a bound of the form
$|S_{N}(x;\alpha,\beta)|\ll_{\varepsilon}N^{1/2+\varepsilon}$ uniformly in $x$
and $N$ (Corollary 5.2) for the rational pairs $(\alpha,\beta)$ we consider.

##  2 Representation-Theoretical Preliminaries

In this section we briefly introduce the necessary representation-theoretical
ingredients needed to define (in Section 3) the Jacobi theta function
$\Theta_{f}$ for any sufficiently regular weight function $f$. For further
details, we refer the reader to [17].

###  2.1 The Heisenberg Group its Schrödinger Representation

Let $\omega:\mathbb{R}^{2k}\times\mathbb{R}^{2k}\to\mathbb{R}$ be the standard
symplectic form,

$\displaystyle\omega\\!\left(\left(\begin{array}[]{c}\\!{\text{\boldmath$\xi$}}_{1}^{{\color[rgb]{1,1,1}.}}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}^{{\color[rgb]{1,1,1}.}}\\!\end{array}\right),\left(\begin{array}[]{c}\\!{\text{\boldmath$\xi$}}^{\prime}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}^{\prime}_{2}\\!\end{array}\right)\right)={\text{\boldmath$\xi$}}_{1}\cdot{\text{\boldmath$\xi$}}^{\prime}_{2}-{\text{\boldmath$\xi$}}^{\prime}_{1}\cdot{\text{\boldmath$\xi$}}_{2},$
(2.5)

where
${\text{\boldmath$\xi$}}_{1},{\text{\boldmath$\xi$}}_{2},{\text{\boldmath$\xi$}}^{\prime}_{1},{\text{\boldmath$\xi$}}^{\prime}_{2}\in\mathbb{R}^{k}.$
We define the Heisenberg group as
$\mathbb{H}(\mathbb{R}^{k})=\mathbb{R}^{2k}\times\mathbb{R}$ with
multiplication law

$\displaystyle({\text{\boldmath$\xi$}},\zeta)({\text{\boldmath$\xi$}}^{\prime},\zeta^{\prime})=({\text{\boldmath$\xi$}}+{\text{\boldmath$\xi$}}^{\prime},\zeta+\zeta^{\prime}+\tfrac{1}{2}\omega({\text{\boldmath$\xi$}},{\text{\boldmath$\xi$}}^{\prime})).$
(2.6)

Note that $\mathbb{R}^{2k}$ is isomorphic to the quotient of
$\mathbb{H}(\mathbb{R}^{k})$ by its centre. The _Schrödinger representation_
$W$ is a representation of $\mathbb{H}(\mathbb{R}^{k})$ on
$\mathcal{U}(\mathrm{L}^{2}(\mathbb{R}^{k}))$, the group of unitary operators
on $\mathrm{L}^{2}(\mathbb{R}^{k})$. Specifically, using the fact that each
element in $\mathbb{H}(\mathbb{R}^{k})$ can be decomposed as

$\displaystyle\left(\\!\begin{pmatrix}{\text{\boldmath$\xi$}}_{1}\\\
{\text{\boldmath$\xi$}}_{2}\end{pmatrix},\zeta\right)=\left(\\!\begin{pmatrix}{\text{\boldmath$\xi$}}_{1}\\\
\mathbf{0}\end{pmatrix},0\right)\left(\\!\begin{pmatrix}\mathbf{0}\\\
{\text{\boldmath$\xi$}}_{2}\end{pmatrix},0\right)\left(\\!\begin{pmatrix}\mathbf{0}\\\
\mathbf{0}\end{pmatrix},\zeta-\frac{{\text{\boldmath$\xi$}}_{1}\cdot{\text{\boldmath$\xi$}}_{2}}{2}\right),$
(2.7)

we define $W$ via

$\displaystyle\left[W\\!\left(\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!\mathbf{0}\\!\end{smallmatrix}\right),0\right)f\right]({\text{\boldmath$w$}})=\mathrm{e}\\!\left(({\text{\boldmath$\xi$}}_{1}\cdot{\text{\boldmath$w$}}\right)f({\text{\boldmath$w$}}),$
(2.8)
$\displaystyle\left[W\\!\left(\left(\begin{smallmatrix}\\!\mathbf{0}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right),0\right)f\right]({\text{\boldmath$w$}})=f({\text{\boldmath$w$}}-{\text{\boldmath$\xi$}}_{2}),$
(2.9)
$\displaystyle\left[W\\!\left(\left(\begin{smallmatrix}\\!\mathbf{0}\\!\\\
\\!\mathbf{0}\\!\end{smallmatrix}\right),\zeta\right)f\right]({\text{\boldmath$w$}})=\mathrm{e}\\!\left(\zeta\right)f({\text{\boldmath$w$}}),$
(2.10)

where $f\in\mathrm{L}^{2}(\mathbb{R}^{k})$. For further details on this
construction, see Section 1.2 of [13].

###  2.2 $\mathrm{Sp}(k,\mathbb{R})$ and its Projective Shale-Weil
Representation

For any $M\in\mathrm{Sp}(k,\mathbb{R})$ we may define a new representation of
$\mathbb{H}(\mathbb{R}^{k})$ as
$W_{M}({\text{\boldmath$\xi$}},\zeta):=W(M{\text{\boldmath$\xi$}},\zeta)$
where $W$ is the Schrödinger representation defined in (2.8)-(2.10). By the
Stone-von Neumann theorem any such representation is irreducible, and
unitarily equivalent. Therefore, there exists a unitary operator $R(M)$ on
$\mathrm{L}^{2}(\mathbb{R}^{k})$ such that

$\displaystyle
R(M)W({\text{\boldmath$\xi$}},\zeta)R(M)^{-1}=W_{M}({\text{\boldmath$\xi$}},\zeta).$
(2.11)

By Schur’s Lemma the map $M\mapsto R(M)$ is unique up to a phase, that is

$\displaystyle R(M_{1}M_{2})=c(M_{1},M_{2})R(M_{1})R(M_{2})$ (2.12)

where $|c(M_{1},M_{2})|=1$. The map
$R:\mathrm{Sp}(k,\mathbb{R})\to\mathcal{U}(\mathrm{L}^{2}(\mathbb{R}^{k}))$ is
therefore a projective unitary representation of $\mathrm{SL}(2,\mathbb{R})$,
known as the _projective Shale-Weil representation_. The phase
$c(M_{1},M_{2})$ can be computed explicitly (see Theorem 1.6.11 in [13]), but
we shall not need it in what follows.

###  2.3 The Jacobi Group and its Projective Schrödinger-Weil Representation

We define the Jacobi group as
$\mathrm{Sp}(k,\mathbb{R})\ltimes\mathbb{H}(\mathbb{R}^{k})$ whose group law
is

$\displaystyle(M;{\text{\boldmath$\xi$}},\zeta)(M^{\prime};{\text{\boldmath$\xi$}}^{\prime},\zeta^{\prime})=(MM^{\prime};{\text{\boldmath$\xi$}}+M{\text{\boldmath$\xi$}}^{\prime},\zeta+\zeta^{\prime}-\tfrac{1}{2}\omega({\text{\boldmath$\xi$}},M{\text{\boldmath$\xi$}}^{\prime})),$
(2.13)

where $\omega$ is as in (2.5). Using the Schrödinger representation $W$ of
$\mathbb{H}(\mathbb{R}^{k})$ from Section 2.1 and the projective Shale-Weil
representation $R$ of $\mathrm{SL}(2,\mathbb{R})$ from Section 2.2, we can
define the _projective Schrödinger-Weil representation_ of the Jacobi group as

$\displaystyle
R(M;{\text{\boldmath$\xi$}},\zeta):=W({\text{\boldmath$\xi$}},\zeta)R(M).$
(2.14)

##  3 The Jacobi Theta Function

Observe that we may embed $\mathrm{SL}(2,\mathbb{R})$ in
$\mathrm{Sp}(k,\mathbb{R})$ via $\bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}\mapsto\bigr{(}\begin{smallmatrix}aI_{k}&bI_{k}\\\
cI_{k}&dI_{k}\end{smallmatrix}\bigr{)}$ where $I_{k}$ is the $k\times k$
identity matrix. Given a function $f:\mathbb{R}^{k}\to\mathbb{R}$ we define
(up to a phase) a theta function
$\Theta_{f}:\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{H}(\mathbb{R}^{k})\to\mathbb{C}$
given by

$\displaystyle\Theta_{f}(M;{\text{\boldmath$\xi$}},\zeta):=\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}[R(M;{\text{\boldmath$\xi$}},\zeta)f]({\text{\boldmath$n$}}),$
(3.1)

provided the series in (3.1) converges absolutely (we shall make sufficient
assumptions for this to hold, see Section 3.2). The function is defined up to
a phase because $R(M)$ in (2.11), and therefore
$R(M;{\text{\boldmath$\xi$}},\zeta)$ in (2.14), are defined up to a phase. To
properly define $\Theta_{f}$, one can pass to the universal cover
$\widetilde{\mathrm{SL}}(2,\mathbb{R})\ltimes\mathbb{H}(\mathbb{R}^{k})$ as
done in [14], [18], [4], [5] when $k=1$. However, since our aim is to study
$|\Theta_{f}|$, this definition of $\Theta_{f}$ will suffice. Let us see that
$|\Theta_{f}|$ may be viewed as a real-valued function on the group
$G:=\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2k}$, with group law

$\displaystyle(M;{\text{\boldmath$\xi$}})(M^{\prime};{\text{\boldmath$\xi$}}^{\prime})=(MM^{\prime};{\text{\boldmath$\xi$}}+M{\text{\boldmath$\xi$}}^{\prime}).$
(3.2)

In fact, by $\eqref{schrodinger-weil-repn-defn}$, we see that
$R(M;{\text{\boldmath$\xi$}},\zeta)=W(\mathbf{0},\zeta)W({\text{\boldmath$\xi$}},0)R(M)$.
Therefore, by (2.10) and (3.1), it follows that

$\displaystyle|\Theta_{f}(M;{\text{\boldmath$\xi$}},\zeta)|=\left|e(\zeta)\sum_{n\in\mathbb{Z}^{k}}R(M;{\text{\boldmath$\xi$}},0)f({\text{\boldmath$n$}})\right|=|\Theta_{f}(M;{\text{\boldmath$\xi$}},0)|,$
(3.3)

and so we may define $|\Theta_{f}|:G\to\mathbb{R}$ as

$\displaystyle|\Theta_{f}|(M;{\text{\boldmath$\xi$}}):=|\Theta_{f}(M;{\text{\boldmath$\xi$}},0)|.$
(3.4)

###  3.1 Writing $|\Theta_{f}|$ in coordinates on
$\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2k}$

Through the embedding of $\mathrm{SL}(2,\mathbb{R})$ in
$\mathrm{Sp(k,\mathbb{R})}$, we obtain an action of
$\mathrm{SL}(2,\mathbb{R})$ on $\mathbb{R}^{2k}$ coming from the group law
(3.2) of $\mathrm{Sp}(k,\mathbb{R})\ltimes\mathbb{R}^{2k}$, namely

$\displaystyle\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\cdot\left(\begin{array}[]{c}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{array}\right)=\left(\begin{array}[]{c}\\!a{\text{\boldmath$\xi$}}_{1}+b{\text{\boldmath$\xi$}}_{2}\\!\\\
\\!c{\text{\boldmath$\xi$}}_{1}+d{\text{\boldmath$\xi$}}_{2}\\!\end{array}\right).$
(3.9)

By Iwasawa decomposition, any $M\in\mathrm{SL}(2,\mathbb{R})$ may be written
as

$\displaystyle M=n_{x}a_{y}k_{\phi}$ (3.10)

where $x+iy\in\mathfrak{H}:=\\{z\in\mathbb{C}:\operatorname{Im}(z)>0\\}$,
$\phi\in[0,2\pi)$, and

$\displaystyle n_{x}:=\left(\begin{matrix}1&x\\\
0&1\end{matrix}\right),\>\>\>\>a_{y}:=\left(\begin{matrix}y^{1/2}&0\\\
0&y^{-1/2}\end{matrix}\right),\>\>\>\>k_{\phi}:=\left(\begin{matrix}\cos\phi&-\sin\phi\\\
\sin\phi&\cos\phi\end{matrix}\right).$ (3.11)

It can be shown (see, e.g., Section 1.6 of [13]) that

$\displaystyle[R(n_{x})f]({\text{\boldmath$w$}})$
$\displaystyle=\mathrm{e}\\!\left(\tfrac{1}{2}\|{\text{\boldmath$w$}}\|^{2}x\right)f({\text{\boldmath$w$}}),$
(3.12) $\displaystyle[R(a_{y})f]({\text{\boldmath$w$}})$
$\displaystyle=y^{\frac{k}{4}}f\\!\left(y^{\frac{1}{2}}{\text{\boldmath$w$}}\right),$
(3.13) $\displaystyle[R(k_{\phi})f]({\text{\boldmath$w$}})$
$\displaystyle=\begin{dcases*}f({\text{\boldmath$w$}})&if
$\phi=0\\!\\!\\!\pmod{2\pi}$,\\\ f(-{\text{\boldmath$w$}})&if
$\phi=\pi\\!\\!\\!\pmod{2\pi}$,\\\
|\sin\phi|^{-\frac{k}{2}}\int_{\mathbb{R}^{k}}\mathrm{e}\\!\left(\tfrac{\frac{1}{2}(\|{\text{\boldmath$w$}}\|^{2}+\|{\text{\boldmath$v$}}\|^{2})\cos\phi-{\text{\boldmath$w$}}\cdot{\text{\boldmath$v$}}}{\sin\phi}\right)f({\text{\boldmath$v$}})\>\mathrm{d}{\text{\boldmath$v$}}&if
$\phi\neq 0\\!\\!\\!\pmod{\pi}$.\end{dcases*}$ (3.14)

In view of (3.10), formulæ (3.12)–(3.14) are enough to define the projective
Shale-Weil representation restricted to $\mathrm{SL}(2,\mathbb{R})$. Define

$\displaystyle f_{\phi}:=R(k_{\phi})f.$ (3.15)

Identifying
$(M,{\text{\boldmath$\xi$}})\in\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2k}$
with
$(z,\phi;{\text{\boldmath$\xi$}})\in\mathfrak{H}\times[0,2\pi)\times\mathbb{R}^{2k}$
via (3.10), the modulus of the theta function $|\Theta_{f}|$ may be written in
‘Iwasawa coordinates’ as

$\displaystyle|\Theta_{f}(z,\phi;{\text{\boldmath$\xi$}})|=y^{k/4}\left|\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}f_{\phi}(({\text{\boldmath$n$}}-{\text{\boldmath$\xi$}}_{2})y^{1/2})\mathrm{e}\\!\left(\tfrac{1}{2}\|{\text{\boldmath$n$}}-{\text{\boldmath$\xi$}}_{2}\|^{2}x+n{\text{\boldmath$\xi$}}_{1}\right)\right|,$
(3.16)

We now see how the generalised quadratic Weyl sums (1.1) can be related to
certain values of $|\Theta_{f}|$. By choosing
$(z,\phi;{\text{\boldmath$\xi$}})=(x+iN^{-2},0;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}+{\text{\boldmath$\beta$}}x\\!\\\
\\!0\\!\end{smallmatrix}\right))$ we have

$\displaystyle|\Theta_{f}(x+\tfrac{i}{N^{2}},0;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}+{\text{\boldmath$\beta$}}x\\!\\\
\\!0\\!\end{smallmatrix}\right))|$
$\displaystyle=N^{-k/2}\\!\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}f(\tfrac{1}{N}{\text{\boldmath$n$}})\,\mathrm{e}\\!\left((\tfrac{1}{2}\|{\text{\boldmath$n$}}\|^{2}+{\text{\boldmath$\beta$}}\cdot{\text{\boldmath$n$}})x+{\text{\boldmath$\alpha$}}\cdot{\text{\boldmath$n$}}\right)$
$\displaystyle=N^{-k/2}S^{f}_{N}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}}).$
(3.17)

Hence, in light of (3.17), bounds of the form
$|S_{N}^{f}(x,{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})|\ll
N^{k/2}$ can be obtained by bounds
$|\Theta_{f}(x+\tfrac{i}{N^{2}},0;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}+{\text{\boldmath$\beta$}}x\\!\\\
\\!0\\!\end{smallmatrix}\right))|\ll 1$.

###  3.2 Regular Cut-off Functions

When we introduced $S_{N}^{f}$ in (1.1), we assumed that $f$ is a Schwartz
function. We now relax this assumption. It is apparent from (3.16) that
$|\Theta_{f}|:\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2k}\to\mathbb{R}$ is
not necessarily well defined pointwise for arbitrary $f$. For instance, as
pointed out in Section 2.6 of [4], taking $k=1$ and $f=\mathds{1}_{(0,1]}$ we
see that $\Theta_{f}(z,0;{\text{\boldmath$\xi$}},\zeta)$ converges since it is
a finite sum, but $f_{\pi/2}(w)$ decays too slowly as $|w|\to\infty$, and
hence the series defining $\Theta_{f}(z,\pi/2;{\text{\boldmath$\xi$}},\zeta)$
does not converge absolutely. Define

$\displaystyle\kappa_{\eta}(f):=\sup_{{\text{\boldmath$w$}},\phi}(1+\|{\text{\boldmath$w$}}\|^{2})^{\eta/2}|f_{\phi}({\text{\boldmath$w$}})|,$
(3.18)

and consider the function space

$\displaystyle\mathcal{S}_{\eta}(\mathbb{R}^{k}):=\\{f:\mathbb{R}^{k}\to\mathbb{R}:\kappa_{\eta}(f)<\infty\\}.$
(3.19)

We say that a cut-off function $f:\mathbb{R}^{k}\to\mathbb{R}$ is _regular_ if
it belongs to $\mathcal{S}_{\eta}(\mathbb{R}^{k})$ for some $\eta>k$. Note
that the regularity of $f$ is sufficient to guarantee that the series (3.16)
defining $|\Theta_{f}|$ is absolutely convergent for every
$(z,\phi;{\text{\boldmath$\xi$}})\in\mathfrak{H}\times[0,2\pi)\times\mathbb{R}^{2k}$.
Regular cut-off functions generalise Schwartz functions. In fact, it can be
shown that Schwartz functions belong to $\mathcal{S}_{\eta}(\mathbb{R}^{k})$
for every $\eta>1$, see Lemma 4.3 in [17].

The following lemma will be useful in the proof of the main theorem.

######  Lemma 3.1.

Let $h:=\left(\left(\begin{smallmatrix}\\!{\text{\boldmath$a$}}\\!\\\
\\!{\text{\boldmath$b$}}\\!\end{smallmatrix}\right),c\right)\in\mathbb{H}(\mathbb{R}^{k})$.
If $f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$ for some $\eta>1$ then
$W(h)f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$. More precisely,

$\displaystyle\kappa_{\eta}(W(h)f)\leq
3^{\eta}(1+\|{\text{\boldmath$a$}}\|^{2}+\|{\text{\boldmath$b$}}\|^{2})^{\eta/2}\,\kappa_{\eta}(f).$
(3.20)

###### Proof.

Due to (2.7) and (2.10), it is enough to consider
$h=(\left(\begin{smallmatrix}\\!{\text{\boldmath$a$}}\\!\\\
\\!\mathbf{0}\\!\end{smallmatrix}\right),0)(\left(\begin{smallmatrix}\\!\mathbf{0}\\!\\\
\\!{\text{\boldmath$b$}}\\!\end{smallmatrix}\right),0)$. Using (2.11) and the
fact that $W_{k_{\phi}}(h)=\
W\\!\left(\left(\begin{smallmatrix}\\!{\text{\boldmath$a$}}\cos\phi-{\text{\boldmath$b$}}\sin\phi\\!\\\
\\!{\text{\boldmath$a$}}\sin\phi+{\text{\boldmath$b$}}\cos\phi\\!\end{smallmatrix}\right),\tfrac{{\text{\boldmath$a$}}\cdot{\text{\boldmath$b$}}}{2}\right)$,
we get

$\displaystyle|[W(h)f]_{\phi}({\text{\boldmath$w$}})|=\left|\left[R(k_{\phi})W\\!\left(\left(\begin{smallmatrix}\\!{\text{\boldmath$a$}}\\!\\\
\\!{\text{\boldmath$b$}}\\!\end{smallmatrix}\right),0\right)f\right]\\!({\text{\boldmath$w$}})\right|$
$\displaystyle=\left|W\\!\left(\left(\begin{smallmatrix}\\!{\text{\boldmath$a$}}\cos\phi-{\text{\boldmath$b$}}\sin\phi\\!\\\
\\!{\text{\boldmath$a$}}\sin\phi+{\text{\boldmath$b$}}\cos\phi\\!\end{smallmatrix}\right),0\right)\cdot
f_{\phi}({\text{\boldmath$w$}})\right|$ (3.21)
$\displaystyle=|f_{\phi}({\text{\boldmath$w$}}-({\text{\boldmath$a$}}\cos\phi+{\text{\boldmath$b$}}\sin\phi))|.$
(3.22)

Now, note that

$\displaystyle\kappa_{\eta}(h\cdot f)$
$\displaystyle=\sup_{{\text{\boldmath$w$}},\phi}\>(1+\|{\text{\boldmath$w$}}\|^{2})^{\eta/2}|f_{\phi}({\text{\boldmath$w$}}-({\text{\boldmath$a$}}\cos\phi+{\text{\boldmath$b$}}\sin\phi))|$
(3.23)
$\displaystyle=\sup_{{\text{\boldmath$w$}},\phi}\>(1+\|{\text{\boldmath$w$}}+({\text{\boldmath$a$}}\cos\phi+{\text{\boldmath$b$}}\sin\phi)\|^{2})^{\eta/2}|f_{\phi}({\text{\boldmath$w$}})|.$
(3.24)

Applying the triangle inequality and the AM-GM inequality to the multiplier in
(3.24) we have

$\displaystyle\sup_{{\text{\boldmath$w$}},\phi}\>(1+\|{\text{\boldmath$w$}}+({\text{\boldmath$a$}}\cos\phi+{\text{\boldmath$b$}}\sin\phi)\|^{2})^{\eta/2}|f_{\phi}({\text{\boldmath$w$}})|$
(3.25) $\displaystyle\leq
3^{\eta}\sup_{{\text{\boldmath$w$}},\phi}\>((1+\|{\text{\boldmath$a$}}\|^{2}+\|{\text{\boldmath$b$}}\|^{2})+\|{\text{\boldmath$w$}}\|^{2})^{\eta/2}|f_{\phi}({\text{\boldmath$w$}})|$
(3.26) $\displaystyle\leq
3^{\eta}(1+\|{\text{\boldmath$a$}}\|^{2}+\|{\text{\boldmath$b$}}\|^{2})^{\eta/2}\kappa_{\eta}(f),$
(3.27)

which proves the lemma. ∎

###  3.3 Invariance Properties

Using (3.10) to parameterise $\mathrm{SL}(2,\mathbb{R})$ by
$\mathfrak{H}\times[0,2\pi)$, we obtain the following transitive action of
$\mathrm{SL}(2,\mathbb{R})$ on $\mathfrak{H}\times[0,2\pi)$:

$\displaystyle\bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}.(z,\phi)=\left(\frac{az+b}{cz+d},\phi+\arg(cz+d)\right).$
(3.28)

We then define an action of special affine linear group $G$ with group law on
$\mathfrak{H}\times[0,2\pi)\times\mathbb{R}^{2k}$ as

$\displaystyle\left(\bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)};{\text{\boldmath$v$}}\right)\\!.(z,\phi;{\text{\boldmath$\xi$}})=\left(\frac{az+b}{cz+d},\phi+\arg(cz+d);{\text{\boldmath$v$}}+\bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}\cdot{\text{\boldmath$\xi$}}\right).$ (3.29)

where $\bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}\cdot{\text{\boldmath$\xi$}}$ is as in (3.9) The
following transformation formulæ for $\Theta_{f}$ follow from Poisson
summation and the definition of $\Theta_{f}$.

######  Theorem 3.2 ([17], “Jacobi 1-3”).

Let $f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$, with $\eta>k$. Let $\Delta<G$ be
the group generated by

$\displaystyle\rho_{1}:=\left(\bigr{(}\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\bigr{)};\mathbf{0}\right),\>\>\rho_{2}:=\left(\bigr{(}\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\bigr{)};\left(\begin{smallmatrix}\\!{\text{\boldmath$s$}}\\!\\\
\\!\mathbf{0}\\!\end{smallmatrix}\right)\right),\>\>\rho_{{\text{\boldmath$m$}},{\text{\boldmath$n$}}}:=\left(I;\left(\begin{smallmatrix}\\!{\text{\boldmath$m$}}\\!\\\
\\!{\text{\boldmath$n$}}\\!\end{smallmatrix}\right)\right),$ (3.30)

where ${\text{\boldmath$s$}}=(\tfrac{1}{2},\dots,\tfrac{1}{2})^{\top}$, and
${\text{\boldmath$m$}},{\text{\boldmath$n$}}\in\mathbb{Z}^{k}$. Then

$\displaystyle|\Theta_{f}(\rho\cdot(z,\phi;{\text{\boldmath$\xi$}}))|=|\Theta_{f}(z,\phi;{\text{\boldmath$\xi$}})|$
(3.31)

for any $\rho\in\Delta$.

It follows that for $f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$ with $\eta>1$,
the function $|\Theta_{f}|$ is a well-defined real-valued function on the
group $\Delta\backslash G$. For convenience, we pass to a finite index
subgroup $\Gamma$ of $\Delta$, whose generators are

$\displaystyle\gamma_{1}=\rho_{1},\>\gamma_{2}=\rho^{2}_{1}\rho_{{\text{\boldmath$u$}},\mathbf{0}}=\left(\bigr{(}\begin{smallmatrix}1&2\\\
0&1\end{smallmatrix}\bigr{)};\mathbf{0}\right),\>\gamma_{{\text{\boldmath$m$}},{\text{\boldmath$n$}}}=\rho_{{\text{\boldmath$m$}},{\text{\boldmath$n$}}},$
(3.32)

where ${\text{\boldmath$u$}}=(-1,\dots,-1)^{\top}$, and
${\text{\boldmath$m$}},{\text{\boldmath$n$}}\in\mathbb{Z}^{k}$.

Clearly, $\Gamma=\Gamma_{\theta}\ltimes\mathbb{Z}^{2k}$, where the lattice

$\displaystyle\Gamma_{\theta}=\langle\bigr{(}\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\bigr{)},\bigr{(}\begin{smallmatrix}1&2\\\
0&1\end{smallmatrix}\bigr{)}\rangle<\mathrm{SL}(2,\mathbb{Z})$ (3.33)

is the so-called _theta group_. We therefore view $|\Theta_{f}|$ as a function
on $\Gamma\backslash G$ or
$\Gamma\backslash(\mathfrak{H}\times[0,2\pi)\times\mathbb{R}^{2k})$, whichever
is more convenient. This quotient can be shown to be non-compact and of with
finite volume according to the Haar measure on $G$.

###  3.4 Fundamental Domains

As the action of $G$ on the space
$\mathfrak{H}\times[0,2\pi)\times\mathbb{R}^{2k}$ defined in (3.29) is
transitive, we may identify $\Gamma\backslash G$ and
$\Gamma\backslash(\mathfrak{H}\times[0,2\pi)\times\mathbb{R}^{2k})$.

######  Proposition 3.3.

A fundamental domain for the action of $\Gamma$ on
$\mathfrak{H}\times[0,2\pi)\times\mathbb{R}^{2k}$ is given by

$\displaystyle\mathcal{F}_{\Gamma}=\mathcal{F}_{\Gamma_{\theta}}\times[0,\pi)\times[0,1)^{2k},$
(3.34)

where
$\mathcal{F}_{\Gamma_{\theta}}=\\{z\in\mathfrak{H}:|z|>1,|z-2|>1,0\leq\operatorname{Re}z<2\\}$
is a fundamental domain for the action of $\Gamma_{\theta}$ on $\mathfrak{H}$.

###### Proof.

The proof is essentially identical to the proof of Proposition 2.5.1 in [5]. ∎

The choice of fundamental domain $\mathcal{F}_{\Gamma_{\theta}}$ has two
cusps, one at $i\infty$ of width two and another at $1$ of width one. For
reasons that will become apparent later, we split the fundamental domain
$\mathcal{F}_{\Gamma}$ into $\mathcal{F}_{\Gamma}^{(\infty)}$ and
$\mathcal{F}_{\Gamma}^{(1)}$ in order to isolate these cusps, namely

$\displaystyle\mathcal{F}_{\Gamma}^{(\infty)}:=\\{(z,\phi;{\text{\boldmath$\xi$}},\zeta)\in\mathcal{F}_{\Gamma}:|z|\geq
1\\},$ (3.35)
$\displaystyle\mathcal{F}_{\Gamma}^{(1)}:=\\{(z,\phi;{\text{\boldmath$\xi$}},\zeta)\in\mathcal{F}_{\Gamma}:|z|<1\\}.$
(3.36)

##  4 Uniform Bounds

###  4.1 Uniform Bounds for fixed $\xi$

We begin with a lemma along the lines of Lemma 2.1 in [4]. Define, for
${\text{\boldmath$v$}}\in\mathbb{R}^{k}$,

$\displaystyle\theta_{k}({\text{\boldmath$v$}}):=\min_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}\|{\text{\boldmath$n$}}-{\text{\boldmath$v$}}\|,$
(4.1)

i.e. Euclidean distance to the closest integer point.

######  Lemma 4.1.

Suppose
${\text{\boldmath$\xi$}}=\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\in\mathbb{R}^{2k}$
is such that $\theta_{k}({\text{\boldmath$\xi$}}_{2})>0$. Let
$f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$ with $\eta>k$. Then there exits a
constant $C=C(k,{\text{\boldmath$\xi$}}_{2},\eta)>0$ such that

$\displaystyle|\Theta_{f}(x+iy,\phi,{\text{\boldmath$\xi$}})|\leq
C\,\kappa_{\eta}(f).$ (4.2)

for every $y\geq\tfrac{1}{2}$, and every $x,\phi$ and
${\text{\boldmath$\xi$}}_{1}$.

###### Proof.

As ${\text{\boldmath$\xi$}}_{2}\not\in\mathbb{Z}^{k}$ it follows that

$\displaystyle|\Theta_{f}(x+iy,\phi,{\text{\boldmath$\xi$}})|$
$\displaystyle\leq
y^{k/4}\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}|f_{\phi}(({\text{\boldmath$n$}}-{\text{\boldmath$\xi$}}_{2})y^{1/2})|$
(4.3)
$\displaystyle\leq\kappa_{\eta}(f)\,y^{k/4}\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}\frac{1}{(1+\|{\text{\boldmath$n$}}-{\text{\boldmath$\xi$}}_{2}\|^{2}y)^{\eta/2}}$
(4.4)
$\displaystyle\leq\kappa_{\eta}(f)\,y^{-\frac{2\eta-k}{4}}\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}\frac{1}{\|{\text{\boldmath$n$}}-{\text{\boldmath$\xi$}}_{2}\|^{\eta}}.$
(4.5)

As $y\geq\tfrac{1}{2}$ it follows that $y^{-\frac{2\eta-k}{4}}\leq
2^{\frac{2\eta-k}{4}}$ since $\eta>k$. Let

$\displaystyle
C(k,{\text{\boldmath$\xi$}}_{2},\eta):=2^{\frac{2\eta-k}{4}}\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}\frac{1}{\|{\text{\boldmath$n$}}-{\text{\boldmath$\xi$}}_{2}\|^{\eta}},$
(4.6)

which is a convergent series for $\eta>k$, as
$\theta_{k}({\text{\boldmath$\xi$}}_{2})>0$.

∎

######  Remark 4.2.

Let
$\mathcal{M}:=\\{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}:|{\text{\boldmath$n$}}-{\text{\boldmath$\xi$}}_{2}|=\theta_{k}({\text{\boldmath$\xi$}}_{2})\\}$.
Note that $1\leq|\mathcal{M}|\leq 2^{k}$. The series

$\displaystyle\sum_{{\text{\boldmath$n$}}\in\mathbb{Z}^{k}}\frac{1}{\|{\text{\boldmath$n$}}-{\text{\boldmath$\xi$}}_{2}\|^{\eta}}=\frac{|\mathcal{M}|}{\theta_{k}({\text{\boldmath$\xi$}}_{2})^{\eta}}+O_{\eta}(1),$
(4.7)

and so the closer ${\text{\boldmath$\xi$}}_{2}$ is to an integer, the larger
the constant $C(k,{\text{\boldmath$\xi$}}_{2},\eta)$ in (4.6) becomes.

######  Corollary 4.3.

Suppose that
$(x+iy;{\text{\boldmath$\xi$}})\in\mathcal{F}_{\Gamma}^{(\infty)}$ where
${\text{\boldmath$\xi$}}=\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\in\mathbb{R}^{2k}$
is such that $\theta_{k}({\text{\boldmath$\xi$}}_{2})>0$. Let
$f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$ with $\eta>k$. Then

$\displaystyle|\Theta_{f}(x+iy,\phi,{\text{\boldmath$\xi$}})|\leq
C(k,{\text{\boldmath$\xi$}}_{2},\eta)\,\kappa_{\eta}(f),$ (4.8)

where $C$ is as in Lemma 4.1.

###### Proof.

This is immediate from Lemma 4.1 since $y\geq\tfrac{1}{2}$ for any
$(x+iy;{\text{\boldmath$\xi$}})\in\mathcal{F}_{\Gamma}^{(\infty)}$. ∎

######  Corollary 4.4.

Suppose $(x+iy;{\text{\boldmath$\xi$}})\in\mathcal{F}_{\Gamma}^{(1)}$ and
${\text{\boldmath$\xi$}}=\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\in\mathbb{R}^{2k}$.
Let $s$ be as in Theorem 3.2, define
$\hat{\text{\boldmath$\xi$}}_{2}:={\text{\boldmath$\xi$}}_{2}-{\text{\boldmath$\xi$}}_{1}+{\text{\boldmath$s$}}$,
and suppose $\theta_{k}(\hat{\text{\boldmath$\xi$}}_{2})>0$. Let
$f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$ with $\eta>k$. Then

$\displaystyle|\Theta_{f}(x+iy,\phi,{\text{\boldmath$\xi$}})|\leq
C(k,\hat{\text{\boldmath$\xi$}}_{2},\eta)\,\kappa_{\eta}(f),$ (4.9)

where $C$ is as in Lemma 4.1.

###### Proof.

Let $\left(x+iy;\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\right)\in\mathcal{F}_{\Gamma}^{(1)}$.
Define $\rho:=\left(\bigr{(}\begin{smallmatrix}0&1\\\
-1&1\end{smallmatrix}\bigr{)};\left(\begin{smallmatrix}\\!\mathbf{0}\\!\\\
\\!{\text{\boldmath$s$}}\\!\end{smallmatrix}\right)\right)=\rho_{2}^{-1}\rho_{1}^{-1}$
and set

$\displaystyle\left(x^{\prime}+iy^{\prime};\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}^{\prime}\\!\\\
\\!{\text{\boldmath$\xi$}}^{\prime}_{2}\\!\end{smallmatrix}\right)\right)=\rho\cdot\left(x+iy;\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\right).$ (4.10)

A computation gives that $y^{\prime}\geq\frac{1}{2}$ and
${\text{\boldmath$\xi$}}^{\prime}_{2}=\hat{\text{\boldmath$\xi$}}_{2}$. Since
$\hat{\text{\boldmath$\xi$}}_{2}\not\in\mathbb{Z}^{k}$, it follows from Lemma
4.1 that

$\displaystyle\left|\Theta_{f}\\!\left(x^{\prime}+iy^{\prime};\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}^{\prime}\\!\\\
\\!{\text{\boldmath$\xi$}}^{\prime}_{2}\\!\end{smallmatrix}\right)\right)\right|\leq
C(k,\hat{\text{\boldmath$\xi$}}_{2},\eta)\,\kappa_{\eta}(f).$ (4.11)

As $|\Theta_{f}|$ is invariant under $\rho\in\Delta$ (see Theorem 3.2), using
(4.10) we obtain (4.9) from (4.11). ∎

###  4.2 Bounds over $\Gamma$-orbits

Recall that $\Gamma=\Gamma_{\theta}\ltimes\mathbb{Z}^{2k}$ acts on
$\mathbb{R}^{2k}$ via the group law as follows:

$\displaystyle\left(\bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)};\left(\begin{smallmatrix}\\!{\text{\boldmath$m$}}\\!\\\
\\!{\text{\boldmath$n$}}\\!\end{smallmatrix}\right)\right)\cdot\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)=\left(\begin{smallmatrix}\\!a{\text{\boldmath$\xi$}}_{1}+b{\text{\boldmath$\xi$}}_{2}\\!\\\
\\!c{\text{\boldmath$\xi$}}_{1}+d{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)+\left(\begin{smallmatrix}\\!{\text{\boldmath$m$}}\\!\\\
\\!{\text{\boldmath$n$}}\\!\end{smallmatrix}\right).$ (4.12)

This descends to an action on the torus $\mathbb{T}^{2k}$, or equivalently on
$[0,1)^{2k}\bmod\mathbb{Z}^{2k}$. Set
$\mathrm{Orb}_{\Gamma}\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)$ to be the orbit of
$\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\bmod\mathbb{Z}^{2k}$
under the action of $\Gamma$. For example, when $k=1$ we have
$\mathrm{Orb}_{\Gamma}(\frac{1}{6},\frac{1}{6})=\\{(\frac{1}{6},\frac{1}{6}),(\frac{1}{6},\frac{1}{2}),(\frac{1}{6},\frac{5}{6}),(\frac{1}{2},\frac{1}{6}),(\frac{1}{2},\frac{5}{6}),(\frac{5}{6},\frac{1}{6}),(\frac{5}{6},\frac{1}{2}),(\frac{5}{6},\frac{5}{6})\\}$,
see also Figure 3

The following are one-parameter subgroups of $G$, which shall be referred to
as the _geodesic_ and _horocycle_ flows, respectively:

$\displaystyle\Phi$
$\displaystyle:=\left\\{\Phi^{t}:=\left(\bigr{(}\begin{smallmatrix}e^{-t/2}&0\\\
0&e^{t/2}\end{smallmatrix}\bigr{)};\mathbf{0}\right)\\!:\>t\in\mathbb{R}\right\\},$
(4.13) $\displaystyle\Psi$
$\displaystyle:=\left\\{\Psi^{x}:=\left(\bigr{(}\begin{smallmatrix}1&u\\\
0&1\end{smallmatrix}\bigr{)};\mathbf{0}\right)\>:\>x\in\mathbb{R}\right\\}.$
(4.14)

We may rewrite the key relationship between $S^{f}_{N}$ and $\Theta_{f}$ given
in (3.17) using the right action of the geodesic and horocycle flows on the
coset space $\Gamma\backslash G$ as

$\displaystyle
N^{-k/2}|S^{f}_{N}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})|=|\Theta_{f}(x+iN^{-2};\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}+{\text{\boldmath$\beta$}}x\\!\\\
\\!\mathbf{0}\\!\end{smallmatrix}\right))|=\left|\Theta_{f}\\!\left(\Gamma(I;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}+{\text{\boldmath$\beta$}}x\\!\\\
\\!\mathbf{0}\\!\end{smallmatrix}\right))\Psi^{x}\Phi^{2\log
N}\right)\right|.$ (4.15)

This observation first appeared in the works of Marklof [14], [15] and plays a
crucial role in the study of the limiting distribution of theta sums (i.e.
when the weight function $f=\mathds{1}_{(0,1]}$).

Recall the definition of $\theta_{k}$ as in (4.1) and $s$ as in Theorem 3.2.
Define

$\displaystyle
m_{k}({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}}):=\inf\left\\{\theta_{k}({\text{\boldmath$\xi$}}_{2})\land\theta_{k}({\text{\boldmath$\xi$}}_{2}-{\text{\boldmath$\xi$}}_{1}+{\text{\boldmath$s$}}):\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\in\mathrm{Orb}_{\Gamma}\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right)\right\\}.$ (4.16)

######  Theorem 4.5.

Let $f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$ with $\eta>k$. Suppose
$\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right)\in\mathbb{R}^{2k}$
with $m_{k}=m_{k}({\text{\boldmath$\alpha$}},-{\text{\boldmath$\beta$}})>0$.
Then

$\displaystyle|S_{N}^{f}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})|\ll_{k,m_{k},\eta,{\text{\boldmath$\beta$}},f}N^{k/2}$
(4.17)

for every $x\in\mathbb{R}$ and every $N\in\mathbb{N}$.

###### Proof.

We have that

$\displaystyle\Gamma\left(I;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}+{\text{\boldmath$\beta$}}x\\!\\\
\\!0\\!\end{smallmatrix}\right)\right)\Psi^{x}\Phi^{t}=\Gamma\left(I;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!-{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right)\right)\Psi^{x}(I;\left(\begin{smallmatrix}\\!0\\!\\\
\\!{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right))\Phi^{t}=\Gamma\left(I;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!-{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right)\right)\Psi^{x}\Phi^{t}(I;\left(\begin{smallmatrix}\\!\mathbf{0}\\!\\\
\\!e^{-t/2}{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right))$ (4.18)

and so, by (3.1)

$\displaystyle\left|\Theta_{f}\\!\left(\Gamma(I;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}+{\text{\boldmath$\beta$}}x\\!\\\
\\!0\\!\end{smallmatrix}\right))\Psi^{x}\Phi^{2\log
N}\right)\right|=|\Theta_{f_{N}}(\Gamma(I;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!-{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right)\Psi^{x}\Phi^{2\log
N}))|,$ (4.19)

where $f_{N}=W(\left(\begin{smallmatrix}\\!\mathbf{0}\\!\\\
\\!N^{-1}{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right),0)f$. By (4.15)
and (4.19) we have

$\displaystyle
N^{-k/2}|S_{N}^{f}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})|=\left|\Theta_{f_{N}}\\!(x+iN^{-2},0;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!-{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right))\right|.$ (4.20)

Let $\gamma\in\Gamma$ be the unique element such that
$\gamma\cdot(x+iN^{-2},0;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!-{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right))=\left(x^{\prime}+iy^{\prime},\phi^{\prime};\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}^{\prime}\\!\\\
\\!{\text{\boldmath$\beta$}}^{\prime}\\!\end{smallmatrix}\right)\right)\in\mathcal{F}_{\Gamma}$.
Then, by the $\Gamma$-invariance of $|\Theta_{f_{N}}|$, we have

$\displaystyle
N^{-k/2}|S_{N}^{f}(x;\alpha,\beta)|=\left|\Theta_{f_{N}}\\!\left(x^{\prime}+iy^{\prime},\phi^{\prime};\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}^{\prime}\\!\\\
\\!{\text{\boldmath$\beta$}}^{\prime}\\!\end{smallmatrix}\right)\right)\right|.$
(4.21)

Let us apply either Corollary 4.3 or Corollary 4.4 depending on whether
$\left(x^{\prime}+iy^{\prime},\phi^{\prime};\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}^{\prime}\\!\\\
\\!{\text{\boldmath$\beta$}}^{\prime}\\!\end{smallmatrix}\right)\right)$
belongs to $\mathcal{F}_{\Gamma}^{(\infty)}$ or $\mathcal{F}_{\Gamma}^{(1)}$.
In the first case we have

$\displaystyle|\Theta_{f}(x^{\prime}+iy^{\prime},\phi^{\prime};\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}^{\prime}\\!\\\
\\!{\text{\boldmath$\beta$}}^{\prime}\\!\end{smallmatrix}\right))|\leq
C(k,{\text{\boldmath$\beta$}}^{\prime},\eta)\,\kappa_{\eta}(f_{N}),$ (4.22)

while in the second case we have the bound

$\displaystyle|\Theta_{f}(x^{\prime}+iy^{\prime},\phi^{\prime};\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}^{\prime}\\!\\\
\\!{\text{\boldmath$\beta$}}^{\prime}\\!\end{smallmatrix}\right))|\leq
C(k,\hat{{\text{\boldmath$\beta$}}^{\prime}},\eta)\,\kappa_{\eta}(f_{N}),$
(4.23)

where
$\hat{{\text{\boldmath$\beta$}}^{\prime}}={\text{\boldmath$\beta$}}^{\prime}-{\text{\boldmath$\alpha$}}^{\prime}+{\text{\boldmath$s$}}$.
The previous two estimates imply

$\displaystyle|\Theta_{f}(x^{\prime}+iy^{\prime},\phi^{\prime};\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}^{\prime}\\!\\\
\\!{\text{\boldmath$\beta$}}^{\prime}\\!\end{smallmatrix}\right))|\leq
M\,\kappa_{\eta}(f_{N}),$ (4.24)

where

$\displaystyle
M=M(k,m_{k},\eta)=\sup\left\\{\max\\{C(k,{\text{\boldmath$\beta$}}^{\prime},\eta),C(k,\hat{{\text{\boldmath$\beta$}}^{\prime}},\eta)\\}:\>\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}^{\prime}\\!\\\
\\!{\text{\boldmath$\beta$}}^{\prime}\\!\end{smallmatrix}\right)\in\mathrm{Orb}_{\Gamma}\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!-{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right)\right\\}.$ (4.25)

The constant $M$ is finite because of the assumption the
$m_{k}({\text{\boldmath$\alpha$}},-{\text{\boldmath$\beta$}})>0$ (recall the
explicit dependence of $C$ on $\theta_{k}$ given in Remark 4.2 and the
definition of $m_{k}$). In other words, the constant $M$ does not depend on
the group element $\gamma$ we used to translate the point
$(x+iN^{-2},0;\left(\begin{smallmatrix}\\!{\text{\boldmath$\alpha$}}\\!\\\
\\!-{\text{\boldmath$\beta$}}\\!\end{smallmatrix}\right))$ back to the
fundamental domain $\mathcal{F}_{\Gamma}$.

By Lemma 3.1 we have that

$\displaystyle\kappa_{\eta}(f_{N})\leq
3^{\eta}(1+\tfrac{1}{N^{2}}\|{\text{\boldmath$\beta$}}\|^{2})^{\eta/2}\kappa_{\eta}(f).$
(4.26)

Finally, combining (4.21), (4.24)–(4.26), and using the trivial lower bound
$N\geq 1$, we obtain

$\displaystyle|S_{N}^{f}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})|\leq
M\,3^{\eta}\,(1+\|{\text{\boldmath$\beta$}}\|^{2})^{\eta/2}\,\kappa_{\eta}(f)N^{k/2}.$
(4.27)

Note that if $f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$ for some $\eta$ then $f$
also belongs to any $\mathcal{S}_{\eta^{\prime}}(\mathbb{R}^{k})$ with
$\eta^{\prime}\leq\eta$. Therefore (4.27) implies (4.17) with

$\displaystyle\inf_{\eta^{\prime}\in(k,\eta]}\left(M(k,m_{k},\eta^{\prime})\,3^{\eta^{\prime}}\,(1+\|{\text{\boldmath$\beta$}}\|^{2})^{\eta^{\prime}/2}\,\kappa_{\eta^{\prime}}(f)\right)$
(4.28)

as implied constant. ∎

######  Remark 4.6.

If we add the minor assumption that $N\geq\|{\text{\boldmath$\beta$}}\|$ in
Theorem 4.5, then we may remove the explicit $\beta$-dependence of the implied
constant in (4.17). In fact, bounding
$1+\tfrac{1}{N^{2}}\|{\text{\boldmath$\beta$}}\|^{2}$ above by $2$ in (4.26)
instead of using the trivial bound $N\geq 1$, we may replace (4.28) with

$\displaystyle\inf_{\eta^{\prime}\in(k,\eta]}M(k,m_{k},\eta^{\prime})\,3^{\eta^{\prime}}2^{\eta^{\prime}/2}\,\kappa_{\eta^{\prime}}(f).$
(4.29)

Naturally, the implied constant still depends implicitly on
$({\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})$ via
$m_{k}({\text{\boldmath$\alpha$}},-{\text{\boldmath$\beta$}})$.

###  4.3 Study of $\Gamma$-Orbits and Conclusions

Set ${\text{\boldmath$\xi$}}_{1}:=(\xi_{11},\cdots,\xi_{1n})^{\top}$ and
${\text{\boldmath$\xi$}}_{2}:=(\xi_{21},\cdots,\xi_{2n})^{\top}$. Define

$\displaystyle\phi$ $\displaystyle:\mathbb{R}^{2k}\to\mathbb{R}^{2k}$ (4.30)
$\displaystyle\phi$
$\displaystyle:\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\mapsto\begin{pmatrix}\left(\begin{smallmatrix}\\!\xi_{11}\\!\\\
\\!\xi_{21}\\!\end{smallmatrix}\right)\\\ \vdots\\\
\left(\begin{smallmatrix}\\!\xi_{1n}\\!\\\
\\!\xi_{2n}\\!\end{smallmatrix}\right)\end{pmatrix}.$ (4.31)

Then

$\displaystyle\phi\left(\bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}\cdot\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)\right)=\begin{pmatrix}M\left(\begin{smallmatrix}\\!\xi_{11}\\!\\\
\\!\xi_{21}\\!\end{smallmatrix}\right)\\\ \vdots\\\
M\left(\begin{smallmatrix}\\!\xi_{1n}\\!\\\
\\!\xi_{2n}\\!\end{smallmatrix}\right)\end{pmatrix}.$ (4.32)

Therefore
$\mathrm{Orb}_{\Gamma}\\!\left(\begin{smallmatrix}\\!{\text{\boldmath$\xi$}}_{1}\\!\\\
\\!{\text{\boldmath$\xi$}}_{2}\\!\end{smallmatrix}\right)$ is determined by
the orbits of $\left(\begin{smallmatrix}\\!\xi_{1i}\\!\\\
\\!\xi_{2i}\\!\end{smallmatrix}\right)$ under the action of $\Gamma_{\theta}$
on $\mathbb{R}^{2}/\mathbb{Z}^{2}$,
$\mathrm{Orb}_{\Gamma_{\theta}}\\!\left(\begin{smallmatrix}\\!\xi_{1i}\\!\\\
\\!\xi_{2i}\\!\end{smallmatrix}\right)$, $1\leq i\leq n$. Recalling the
definition (4.16), we can use the previous observation to can guarantee that
$m_{k}({\text{\boldmath$\xi$}}_{1},{\text{\boldmath$\xi$}}_{2})>0$ by checking
that $m_{1}(\xi_{1i},\xi_{2i})$ is positive for at least one index $1\leq
i\leq n$. To this end, let
$L_{U}:=\\{\left(\begin{smallmatrix}\\!\xi_{1}\\!\\\
\\!\xi_{2}\\!\end{smallmatrix}\right)\in\mathbb{R}^{2}:\xi_{2}=0\\}$, and
$L^{\pm}_{V}:=\\{\left(\begin{smallmatrix}\\!\xi_{1}\\!\\\
\\!\xi_{2}\\!\end{smallmatrix}\right)\in\mathbb{R}^{2}:\xi_{2}=\xi_{1}\pm\tfrac{1}{2}\\}$.
For $(\alpha,\beta)\in\mathbb{R}^{2}$ we define

$\displaystyle U^{(\alpha,\beta)}$
$\displaystyle:=L_{U}\cap\mathrm{Orb}_{\Gamma_{\theta}}\left(\begin{smallmatrix}\\!\alpha\\!\\\
\\!\beta\\!\end{smallmatrix}\right)$ (4.33) $\displaystyle V^{(\alpha,\beta)}$
$\displaystyle:=L_{V}\cap\mathrm{Orb}_{\Gamma_{\theta}}\left(\begin{smallmatrix}\\!\alpha\\!\\\
\\!\beta\\!\end{smallmatrix}\right).$ (4.34)

We have the following

######  Proposition 4.7.

Let $(\alpha,\beta)\in\mathbb{Q}^{2}$ where $\\{\alpha\\}=\tfrac{a}{q}$ and
$\\{\beta\\}=\tfrac{b}{q}$ with $\gcd(a,b,q)=1$ and $q>1$. Then
$|U^{(\alpha,\beta)}|=|V^{(\alpha,\beta)}|=0$ if and only if $a$ and $b$ are
both odd, and $q=2m$ where $m$ is odd.

###### Proof.

This follows immediately from Propositions 3.2.3, 6.2.1 and 6.1.1 in [5]. ∎

Note that
$|\mathrm{Orb}_{\Gamma_{\theta}}\\!\left(\begin{smallmatrix}\\!\alpha\\!\\\
\\!\beta\\!\end{smallmatrix}\right)|<\infty$ for any
$(\alpha,\beta)\in\mathbb{Q}^{2}$. It is important to observe that for
$(\alpha,\beta)\in\mathbb{Q}^{2}$ we have that $m_{1}(\alpha,\beta)>0$ if and
only if the $\Gamma_{\theta}$-orbit of
$\left(\begin{smallmatrix}\\!\alpha\\!\\\ \\!\beta\\!\end{smallmatrix}\right)$
does not intersect neither $L_{U}$ nor $L_{V}^{\pm}$. Hence, Proposition 4.7
gives a classification of all rational pairs $(\alpha,\beta)\in\mathbb{Q}^{2}$
for which $m_{1}(\alpha,\beta)>0$.

The orbit
$\mathrm{Orb}_{\Gamma_{\theta}}\\!\left(\begin{smallmatrix}\\!\alpha\\!\\\
\\!\beta\\!\end{smallmatrix}\right)$ for $(\alpha,\beta)\in\mathbb{Q}^{2}$
also exhibits various symmetries:

######  Proposition 4.8.

For any $(\alpha,\beta)\in\mathbb{Q}^{2}$, we have that

$\displaystyle
m_{1}(\alpha,\beta)=m_{1}(\alpha,-\beta)=m_{1}(-\alpha,\beta)=m_{1}(-\alpha,-\beta).$
(4.35)

###### Proof.

This is immediate from Lemma 3.3.1 in [5]. ∎

We can now state and prove our Main Theorem, which generalises Theorem 1.1
from Schwartz cut-offs to to regular cut-offs. It is basically a version of
Theorem 4.5, where the assumption that
$m_{k}({\text{\boldmath$\alpha$}},-{\text{\boldmath$\beta$}})>0$ is replaced
by a concrete, easy to check, sufficient condition (coming from Proposition
4.7).

######  Theorem 4.9 (Main Theorem, for regular cut-offs).

Let $f\in\mathcal{S}_{\eta}(\mathbb{R}^{k})$ with $\eta>k$. Let
${\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}}\in\mathbb{R}^{k}$, where
${\text{\boldmath$\alpha$}}=(\alpha_{1},\cdots,\alpha_{k})$ and
${\text{\boldmath$\beta$}}=(\beta_{1},\cdots,\beta_{k})$. Suppose that at
least one of $(\alpha_{i},\beta_{i})$ for $1\leq i\leq k$ can be written as
$(\alpha_{i},\beta_{i})=(\tfrac{a}{2m},\tfrac{b}{2m})\in\mathbb{Q}^{2}$ with
$\gcd(a,b,m)=1$ and $a$, $b$, and $m$ all odd. Let $\beta$ the smallest (in
absolute value) of the $\beta_{i}$’s corresponding to such pairs. Then

$\displaystyle|S_{N}^{f}(x;{\text{\boldmath$\alpha$}},{\text{\boldmath$\beta$}})|\ll_{k,m,\eta,\beta,f}N^{k/2}.$
(4.36)

###### Proof.

Suppose that, without loss of generality,
$(\alpha_{1},\beta_{1})=(\tfrac{a}{2m},\tfrac{b}{2m})\in\mathbb{Q}^{2}$ is
such that $\gcd(a,b,m)=1$, and $a$, $b$, and $m$ are all odd. By Propositions
4.7 and 4.8 we are guaranteed that
$m_{1}(\alpha_{1},-\beta_{1})=m_{1}(\alpha_{1},\beta_{1})>0$. Clearly
$m_{k}({\text{\boldmath$\alpha$}},-{\text{\boldmath$\beta$}})\geq
m_{1}(\alpha_{1},\beta_{1})>0$. It is easy to check that
$m_{1}(\alpha_{1},\beta_{1})\geq\frac{1}{2m}$ (see Lemma 5.0.2 in [5]) and so
by Theorem 4.5 we have the result. ∎

###### Proof of Theorem 1.1.

If $f$ is of Schwartz class, then we can apply Theorem 4.9 for every $\eta>k$.
Equivalently, in the case of Scwhartz cut-offs, the dependence upon $\eta$ in
(4.28) —and hence in (4.36)— is removed by taking the infimum over all
$\eta^{\prime}>k$. ∎

######  Remark 4.10.

If $k=1$ then

$\displaystyle C(k,\xi_{2},\eta)$
$\displaystyle=2^{\tfrac{2\eta-1}{4}}\left(\frac{1}{\theta_{1}(\xi_{2})^{\eta}}+\sum_{n=1}^{\infty}\left(\frac{1}{(n-\theta_{1}(\xi_{2}))^{\eta}}+\frac{1}{(n+\theta_{1}(\xi_{2}))^{\eta}}\right)\right),$

and similarly for $C(k,\hat{\xi}_{2},\eta$). Therefore, we may bound
$M(k,m_{1},\eta^{\prime})$ in terms of the Riemann zeta function evaluated at
$\eta^{\prime}$ and in terms of the denominator $2m$ used to uniquely write
$(\alpha,\beta)$ as $\alpha=\frac{a}{2m}$, $\beta=\frac{b}{2m}$ with
$\gcd(a,b,m)=1$. We obtain

$\displaystyle\inf_{\eta^{\prime}\in(1,\eta]}\left(2^{\frac{2\eta^{\prime}-1}{4}}3^{\eta^{\prime}}(2m+2\zeta(\eta^{\prime})(2^{\eta^{\prime}}-1))(1+\beta^{2})^{\eta^{\prime}/2}\kappa_{\eta^{\prime}}(f)\right)$
(4.37)

as an implied constant for (4.36). For Schwartz cut-offs, such as those in
Corollary 1.2, the infimum in (4.37) is taken over all $\eta^{\prime}>1$. See
Section 5.1.

######  Remark 4.11.

Similarly to Remark 4.6, if we make the additional mild assumption that
$N\geq\|{\text{\boldmath$\beta$}}\|$ in Theorem 4.9, then we can remove the
explicit dependence upon $\beta$ in (4.36). In the case where $k=1$ the factor
$(1+\beta^{2})^{\eta^{\prime}/2}$ in (4.37) may then be replaced by
$2^{\eta^{\prime}/2}$. This is done in Theorem 1.4 since $(\alpha,\beta)$ are
assumed to be in the torus $\in\mathbb{R}^{2}/\mathbb{Z}^{2}$.

##  5 Applications

###  5.1 Uniform Bounds for $\vartheta_{3}$

The classical Jacobi theta function $\vartheta_{3}(z,w)$ is defined in (1.4)
for $z,w\in\mathbb{C}$ with $\operatorname{Im}(w)>0$. It is often written as
$q$-series (where $q=e^{-\pi iw}$) as

$\displaystyle\vartheta_{3}(z;q)=\sum_{n\in\mathbb{Z}}e(nz)q^{n^{2}}=1+2\sum_{n=0}^{\infty}\cos(2nz)q^{n^{2}},$
(5.1)

where $|q|<1$. Note that if $\operatorname{Im}(w)\to 0^{+}$, then $|q|\to
1^{-}$. Considering the Schwartz function $f(u)=e^{-\pi u^{2}}$ in (3.17) we
see that

$\displaystyle|\Theta_{f}(x+iy,0;\left(\begin{smallmatrix}\\!\xi_{1}\\!\\\
\\!0\\!\end{smallmatrix}\right))|$
$\displaystyle=y^{1/4}\left|\sum_{n\in\mathbb{Z}}e^{2\pi i\xi_{1}n+\pi
i\left(x+iy\right)n^{2}}\right|$ (5.2)
$\displaystyle=y^{1/4}|\vartheta_{3}(\xi_{1},x+iy)|.$ (5.3)

Taking $y=\varepsilon=N^{-2}$ and $\xi_{1}=\alpha+\beta x$ gives

$\displaystyle\frac{1}{\sqrt{N}}|S_{N}^{f}(x;\alpha,\beta)|=\varepsilon^{\frac{1}{4}}\left|\vartheta_{3}\\!\left(\alpha+\beta
x,x+i\varepsilon\right)\right|.$ (5.4)

We are now ready to prove Theorem 1.4 where the implied constant in (1.5) can
be taken as

$\displaystyle\min_{1<\eta<4/3}\left(2^{-\frac{1}{4}}\,6^{\eta}\,(2m+2\zeta(\eta)(2^{\eta}-1))\right).$
(5.5)

###### Proof of Theorem 1.4.

Recall (3.14)-(3.15). We claim that for $f(u)=e^{-\pi u^{2}}$ and every
$\phi\in\mathbb{R}$ we have

$\displaystyle|f_{\phi}(w)|=e^{-\pi w^{2}}.$ (5.6)

This is obvious if $\phi\equiv 0\>(\bmod\>\pi)$. For all other $\phi$’s, we
first use the identity (see [1] 7.4.32)

$\displaystyle\int_{-\infty}^{\infty}e^{A+Bt+Ct^{2}}\>\mathrm{d}t=\frac{e^{A-\frac{B^{2}}{4C}}\sqrt{\pi}}{\sqrt{-C}}$
(5.7)

for $A,B,C\in\mathbb{C}$ with $\operatorname{Re}(C)<0$. We have

$\displaystyle|f_{\phi}(w)|$
$\displaystyle=\left|\frac{1}{\sqrt{|\sin\phi|}}\int_{-\infty}^{\infty}e\\!\left(\frac{\tfrac{1}{2}(w^{2}+t^{2})\cos\phi+wt}{\sin\phi}\right)e^{-\pi
t^{2}}\>\mathrm{d}t\right|$ (5.8)
$\displaystyle=\left|\frac{\sqrt{\pi}}{\sqrt{|\sin\phi|}}\frac{e^{\pi
iw^{2}\cot\phi-\frac{-2\pi
iw\csc\phi}{4(-\pi+i\pi\cot\phi)}}}{\sqrt{-(-\pi+i\pi\cot\phi)}}\right|=\left|\frac{1}{\sqrt{|\sin\phi|}}\frac{e^{-\pi
w^{2}}}{\sqrt{1-i\cot\phi}}\right|.$ (5.9)

Note that
$|(1-i\cot\phi)^{-1/2}|=|(1-i\cot\phi)(1+i\cot\phi)|^{-1/4}=|\csc^{2}\phi|^{-1/4}=|\csc\phi|^{-1/2}$
and so

$\displaystyle\left|\frac{1}{\sqrt{|\sin\phi|}\sqrt{1-i\cot\phi}}\right|=1,$
(5.10)

yielding (5.6). Now recall and (3.18) let $\eta>1$. We claim that

$\displaystyle\kappa_{\eta}(f)=\begin{dcases*}1&if $1<\eta<2\pi$\\\
e^{\pi}(\tfrac{\eta}{2\pi})^{\eta/2}e^{-\eta/2}&if $\eta\geq
2\pi$.\end{dcases*}$ (5.11)

By (5.6) we have

$\displaystyle\kappa_{\eta}(f)=\sup_{\phi,w}(1+w^{2})^{\eta/2}|f_{\phi}(w)|=\max_{w}(1+w^{2})^{\eta/2}e^{-\pi
w^{2}}.$ (5.12)

By finding the zeroes of the derivative
$\tfrac{\mathrm{d}}{\mathrm{d}w}(1+|w|^{2})^{\eta/2}e^{-\pi w^{2}}=e^{-\pi
w^{2}}(w(w^{2}+1)^{\tfrac{\eta}{2}-1}(\eta-2\pi(w^{2}+1)))$, it is not hard to
see that if $1<\eta\leq 2\pi$, then $\max_{w}(1+w^{2})^{\eta/2}e^{-\pi w^{2}}$
occurs at $w=0$ and equals 1. If $\eta>2\pi$, then
$\max_{w}(1+w^{2})^{\eta/2}e^{-\pi w^{2}}$ occurs at
$w=\pm\sqrt{\tfrac{\eta}{2\pi}-1}$ and equals
$e^{\pi}(\tfrac{\eta}{2})^{\eta/2}e^{-\eta/2}$.

Now let us apply Corollary 1.2 with $f(u)=e^{-\pi u^{2}}$ and
$N=\varepsilon^{-\frac{1}{2}}$. The implied constant in (1.3) can be taken as
$\inf_{\eta>1}h(m,\eta)$, where
$h(m,\eta):=3^{\eta}2^{\eta-\frac{1}{4}}(2m+2\zeta(\eta)(2^{\eta}-1))\kappa_{\eta}(f)$
(see Remarks 4.10-4.11). Using (5.11) and recalling that the $\zeta$-function
$\eta\mapsto\zeta(\eta)$ has a simple pole at $\eta=1$, we see that
$\lim_{\eta\to 1^{+}}h(m,\eta)=\lim_{\eta\to\infty}h(m,\eta)=\infty$ for every
$m\geq 1$ and hence the infimum is a minimum, attained at some
$\eta^{*}(m)>1$. We leave it to the reader to verify that such point of
minimum is unique. Since $m\mapsto h(m,\eta)$ is increasing, we have
$\eta^{*}(m)\leq\eta^{*}(1)$. It is clear that $\eta^{*}(1)\leq 2\pi$ and we
can numerically estimate $\eta^{*}(1)\approx 1.263<\frac{4}{3}$, i.e. when
taking the minimum we can restrict $\eta$ to the interval $(1,\frac{4}{3})$,
where $\kappa_{\eta}(f)=1$, thus obtaining (5.5). ∎

######  Remark 5.1.

Note the bound (1.5) is the same for all rational pairs
$(\frac{a}{2m},\frac{b}{2m})$ with $\gcd(a,b,m)=1$ and $a,b,m$ all odd, since
(5.5) only involves the denominator $2m$. It is easy to see that, given $m$
odd, the number of such pairs in $\mathbb{R}^{2}/\mathbb{Z}^{2}$ is between
$\frac{8m^{2}}{\pi^{2}}$ and $m^{2}$, their exact number being given by the
second Jordan totient function
$J_{2}(m)=m^{2}\prod_{p|m}\left(1-p^{-2}\right)$, see Remark 7.1.2 in [5].
Figure 3 illustrates the uniformity of the bound (1.5) across all rational
pairs in $\mathbb{R}^{2}/\mathbb{Z}^{2}$ with denominator $6$ that satisfy the
hypotheses of Theorem 1.4.

Figure 3: Illustration of Theorem 1.4 for $m=3$. There are $J_{2}(3)=8$
rational pairs $(\alpha,\beta)\in\mathbb{R}^{2}/\mathbb{Z}^{2}$ satisfying
$\alpha=\frac{a}{2m}$, $\beta=\frac{b}{2m}$ with $\gcd(a,b,m)=1$ and $a,b,m$
all odd, see Remark 5.1. The corresponding 8 functions $[0,4]\ni
x\mapsto\varepsilon^{\frac{1}{4}}\left|\vartheta_{3}\\!\left(\alpha+\beta
x,x+i\varepsilon\right)\right|\in\mathbb{R}_{\geq 0}$ are shown for various
values of $\varepsilon\in\\{0.2,0.1,0.05,0.025\\}$. For one of these pairs,
$(\frac{1}{2},\frac{1}{6})$, the function
$x\mapsto\varepsilon^{\frac{1}{4}}\left|\vartheta_{3}\\!\left(\alpha+\beta
x,x+i\varepsilon\right)\right|\in\mathbb{C}$ is shown in Figure 1.

###  5.2 Estimates for $S_{N}$ independent of $x$

Recall that $S_{N}(x;\alpha,\beta)=\sum_{n=1}^{N}e((\tfrac{1}{2}n^{2}+\beta
n)x+\alpha n)$. For fixed $N$ we can find a regular cut-off function
$\chi^{(N)}\in S_{\eta}(\mathbb{R}^{k})$ with $\eta=2$, supported on
$[0,1+\tfrac{1}{N}]$, and $\chi^{(N)}(w)=1$ for $w\in[\tfrac{1}{N},1]$. An
explicit construction is given in [4], equation (3.9). In particular,

$\displaystyle\tfrac{1}{\sqrt{N}}|S_{N}(x;\alpha,\beta)|=|\Theta_{\chi^{(N)}}(x+\tfrac{1}{N^{2}},0;\left(\begin{smallmatrix}\\!\alpha+\beta
x\\!\\\ \\!0\\!\end{smallmatrix}\right))|.$ (5.13)

Proposition 6.5 in [3] shows that

$\displaystyle\kappa_{\eta}(\chi^{(N)})\ll N^{\eta-1}.$ (5.14)

We obtain the following corollary of Main Theorem 4.9.

######  Corollary 5.2.

Suppose $(\alpha,\beta)=(\tfrac{a}{2m},\tfrac{b}{2m})\in\mathbb{Q}^{2}$ where
$\gcd(a,b,m)=1$ and $a,b,$ and $m$ are all odd. Then

$\displaystyle|S_{N}(x;\alpha,\beta)|\ll_{\varepsilon}N^{\tfrac{1}{2}+\varepsilon}$
(5.15)

for every $0<\varepsilon\leq 1$ and every $x,N$.

###### Proof.

Using (5.13), Theorem 4.9 with implied constant (4.37), and (5.14), we obtain

$\displaystyle|S_{N}(x;\alpha,\beta)|$
$\displaystyle\leq\left(2^{\frac{2\eta-1}{4}}3^{\eta}(2q+2\zeta(\eta)(2^{\eta}-1))(1+\beta)^{\eta}\kappa_{\eta}(\chi^{(N)})\right)N^{\tfrac{1}{2}}$
(5.16)
$\displaystyle\ll\left(2^{\frac{2\eta-1}{4}}3^{\eta}(2q+2\zeta(\eta)(2^{\eta}-1))(1+\beta)^{\eta}\right)N^{\tfrac{1}{2}+(\eta-1)}.$
(5.17)

Taking $\varepsilon=\eta-1$ close to $0^{+}$ gives the result. ∎

######  Remark 5.3.

Notice that as $\varepsilon\to 0$, the implied constant in (5.15) goes to
$\infty$ due to the presence of the Riemann $\zeta$-function in the implied
constant.

######  Remark 5.4.

As mentioned in the introduction, this estimate is not optimal. For instance,
it follows from Corollary 1.2 in [8] (cf. also [6],[19]) that there is a full
measure set $\mathcal{A}\subseteq[0,1]$ such that for any $x\in\mathcal{A}$

$\displaystyle|S_{N}(x;\alpha,\beta)|\ll_{x,\varepsilon}\sqrt{N}(\log
N)^{\frac{1}{4}+\varepsilon}.$ (5.18)

for every $N$. For typical $x$, this bound is certainly better than (5.15). On
the other hand, for the class of rational pairs $(\alpha,\beta)$ we consider,
the constant implied in 5.18 does depend on $x$, while the one in (5.15) is
uniform in $x$.

## Acknowledgements

The first author acknowledges the support from the NSERC Discovery Grant
RGPIN-2022-04330. The results presented here were partly developed in the PhD
thesis of the second author. We wish to thank Jens Marklof, Ram M. Murty, and
Brad Rodgers for several fruitful discussions on the subject of this work.

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