# An $L^{4}$ maximal estimate for quadratic Weyl sums

Alex Barron

###### Abstract.

We show that

$\bigg{\|}\sup_{0<t<1}\big{|}\sum_{n=1}^{N}e^{2\pi
i(n(\cdot)+n^{2}t)}\big{|}\bigg{\|}_{L^{4}([0,1])}\leq
C_{\epsilon}N^{3/4+\epsilon}$

and discuss some applications to the theory of large values of Weyl sums. This
estimate is sharp for quadratic Weyl sums, up to the loss of $N^{\epsilon}$.

##  1\. Introduction

In this paper we prove the following maximal estimate for quadratic Weyl sums.

######  Theorem 1.1.

For any $\epsilon>0$ there exists $C_{\epsilon}>0$ such that

(1.1) $\bigg{\|}\sup_{0<t<1}\big{|}\sum_{n=1}^{N}e^{2\pi
i(n(\cdot)+n^{2}t)}\big{|}\bigg{\|}_{L^{4}([0,1])}\leq
C_{\epsilon}N^{3/4+\epsilon}.$

Some applications of this result are discussed below in Section 1.1. The
dependence on $N$ is sharp up to the loss of $N^{\epsilon}$, as can be seen by
observing that the supremum is essentially $N$ for each
$x\in[0,10^{-6}N^{-1}]$. A similar argument shows that we cannot take $p>4$ in
(1.1) without increasing the loss in $N$. Moreover, there exists a set
$E\subset[0,1]$ with $|E|\gtrsim 1$ and $\sup_{t}|\sum_{n=1}^{N}e^{2\pi
i(nx+n^{2}t)}|\gtrsim N^{3/4}$ when $x\in E$ (see [MV] or the end of Section 4
below), and so the loss in $N$ cannot be improved if $p$ is decreased (up to
the factor of $N^{\epsilon}$).

In [MV] Moyua and Vega proved that

(1.2) $\bigg{\|}\sup_{0<t<1}\big{|}\sum_{n=1}^{N}a_{n}e^{2\pi
i(n(\cdot)+n^{2}t)}\big{|}\bigg{\|}_{L^{p}([0,1])}\leq
C_{\epsilon}N^{1/3+\epsilon}\big{(}\sum_{n=1}^{N}|a_{n}|^{2}\big{)}^{1/2},\ \
\ \ 1\leq p\leq 6,$

and when $p=6$ this is sharp up to the loss of $N^{\epsilon}$ (let $a_{n}=1$
and restrict to $[0,10^{-6}N^{-1}]$ as above). Theorem 1.1 improves (1.2) in
the range $1\leq p\leq 4$ in the special case where $a_{n}=1$ for each $n$. It
is conjectured that we have the stronger estimate

(1.3) $\bigg{\|}\sup_{0<t<1}\big{|}\sum_{n=1}^{N}a_{n}e^{2\pi
i(n(\cdot)+n^{2}t)}\big{|}\bigg{\|}_{L^{p}([0,1])}\leq
C_{\epsilon}N^{1/4+\epsilon}\big{(}\sum_{n=1}^{N}|a_{n}|^{2}\big{)}^{1/2},\ \
\ \ 1\leq p\leq 4.$

This conjecture is still open, though Theorem 1.1 implies that it holds in the
Weyl sum case. Maximal estimates of the type (1.1) for more general Weyl sums
were recently considered in [CS4] by Chen and Shparlinski, who studied maximal
analogues of the Vinogradov mean-value estimates proved by Bourgain, Demeter,
and Guth in [BDG] using $\ell^{2}$ decoupling, and by Wooley in [Wo1] using
efficient congruencing.

Maximal estimates of the form (1.2), (1.3) imply pointwise convergence results
for the linear Schrödinger equation on the torus ${\mathbb{T}},$ which we
identify with $[0,1]$. In particular, by using Littlewood-Paley theory and the
Stein maximum principle we see that (1.3) with $p=2$ is equivalent to

$\lim_{t\rightarrow 0}e^{it\Delta_{{\mathbb{T}}}}f(x)=f(x)\text{ a.e. },\ \ \
\ \text{ for all }f\in H^{1/4+}({\mathbb{T}})$

(see for example Appendix A in [P]). The corresponding pointwise convergence
result for the Schrödinger operator on ${\mathbb{R}}$ is known to hold, and
was originally proved by Carleson in [Ca]. Indeed, one has

$\lim_{t\rightarrow 0}e^{it\Delta_{{\mathbb{R}}}}f(x)=f(x)\text{ a.e. },\ \ \
\ f\in H^{1/4}({\mathbb{R}})$

and this is sharp as shown in [DK]. More generally, recent developments in
Fourier restriction theory have led to the near-complete resolution of the
pointwise convergence problem for the linear Schrödinger equation on
${\mathbb{R}}^{n}$, originally posed in [Ca] and [DK]. The problem is to
determine the minimal $s>0$ for which

(1.4)
$\|\sup_{0<t<1}|e^{it\Delta_{{\mathbb{R}}^{n}}}f|\|_{L^{p}(B^{n}(0,1))}\leq
C\|f\|_{H^{s}({\mathbb{R}}^{n})}$

for some $p\geq 1$. In the breakthrough works of Du, Guth, and Li [DGL] and Du
and Zhang [DZ] the estimate (1.4) was proved with sharp Sobolev index (up to
the endpoint) in all dimensions $n\geq 2$ (the case $n=2$ was proved with
$p=3$ in [DGL], and the case $n\geq 3$ was proved with $p=2$ in [DZ]). Their
arguments involve several recent advancements in Fourier restriction theory,
including the polynomial method, refined Strichartz estimates, $\ell^{2}$
decoupling, and a fractal refinement of the broad-narrow reduction of Bourgain
and Guth [BG].

In the case $n=1$ the argument due to Carleson is much simpler than the recent
proofs given in higher dimensions. Unfortunately this argument does not
readily transfer to the case of the torus, since the kernel of the periodic
Schrödinger operator is more erratically behaved than the kernel of the
Euclidean operator. Indeed, in the periodic case the kernel is given by the
quadratic Weyl sum

$w_{N}(x,t)=\sum_{n=1}^{N}e^{2\pi i(nx+n^{2}t)}$

which is known to be large whenever $(x,t)$ is close to a rational point
$(b/q,a/q)$ with $1\leq q<N^{1/2}$. Our main theorem confirms that (1.3) holds
for this kernel, though the methods do not appear to be enough to prove the
conjecture for general solutions.

The proof of Theorem 1.1 is inspired by the argument of Du and Zhang in [DZ].
Three important features of their work play a key role in our argument:
induction on the scale $N$, a version of parabolic rescaling, and the ‘one-
dimensional’ structure of the set where the supremum is attained. However,
there are new complications of a number-theoretic nature that arise.

In order to prove an estimate like (1.1) one hopes to show that the supremum
in $t$ is ‘on average’ no worse than $N^{3/4+\epsilon}$. More specifically, we
already lose $N^{3/4}$ from $x\in[0,cN^{-1}]$ and so the goal is to show that
remaining points do not contribute much more to the $L^{4}$ norm. From
classical Weyl estimates we know that the supremum is larger than $N^{3/4}$
whenever $|x-a/q|<1/(100N)$ for rationals $a/q$ with $1\leq q<N^{1/2}$ and $q$
odd (some conditions on $a$ are required if $q$ is even; see for example
[Oh]). These intervals $I_{a/q}$, which correspond to ‘major arcs’ in the
Hardy-Littlewood circle method, are pairwise-disjoint for $q$ in this range
and can be shown to cover a subset $E\subset{\mathbb{T}}$ of measure
$|E|\gtrsim 1$ (see for example [MV] or the argument at the end of Section 4
below). One must then show that the Weyl sum cannot frequently be larger than
$N^{3/4}$ on $E$, and also that the supremum cannot frequently be larger than
$N^{3/4}$ if $x$ is outside of $E$. The latter is more challenging, since
classical Weyl estimates do not give good information about lower bounds for
the sum if $x$ is outside the union of the $I_{a/q}$. Moreover, the times $t$
at which the supremum is attained will be ‘major arc’ points, and so it is not
clear if it is possible to exploit the cancellation one typically obtains from
the ‘minor arcs’ in the circle method.

To get around these issues we consider a certain collection of axis-parallel
rectangles $Q=I\times J$ in ${\mathbb{T}}^{2}$ where the Weyl sum attains
large values, following the set-up of the argument in [DZ]. The rectangles are
then grouped according to the Diophantine approximation properties of $t\in
J$. If the times $t\in J$ are sufficiently close to a rational $a/q$ with
$q\geq N^{c\epsilon^{2}}$ we show that we can control their contribution by
induction on the scale $N$. On the other hand, if $t\in J$ are close to a
rational $a/q$ with small denominator $q$ we will see we can directly apply a
local-in-time maximal estimate of Moyua and Vega (see Theorem 2.1 below).

There are some similarities between this approach and the broad-narrow
reduction of Bourgain and Guth [BG] which is now a common tool in Fourier
restriction theory and in particular played a key role in [DGL] and [DZ].
However, these connections appear to be only heuristic, and we have not found
a more precise way to bring into play tools from restriction theory. It would
be of interest, for example, to see if it is possible to approach the
conjectured bound (1.3) using some kind of refined $\ell^{2}$ decoupling
estimates (for example similar to those proved in [GIOW]).

###  1.1. An application of Theorem 1.1: Large values of Weyl sums

In Section 3 we will derive a few direct corollaries of our maximal estimate.
For example, let $\mathcal{S}_{\alpha}(N)$ denote the set of
$x\in{\mathbb{T}}$ such that

$\sup_{0<t<1}\bigg{|}\sum_{n=1}^{N}e^{2\pi i(xn+tn^{2})}\bigg{|}\geq
N^{\alpha}.$

We will show in Corollary 3.1 that if $3/4<\alpha\leq 1$ then

(1.5) $|\mathcal{S}_{\alpha}(N)|\lesssim_{\epsilon}N^{3-4\alpha+\epsilon}.$

This result gives us information about pointwise bounds for the Weyl sum
assuming only information about $x$, and should be compared with the classical
estimate in Proposition 4.4 below. The estimate (1.5) should also be compared
with recent work of Chen and Shparlinski, who obtained similar bounds for
exponential sums in [CS]. In the Weyl sum case their estimates in [CS], Lemma
3.2 and Corollary 3.8, show that

$|S_{\alpha}(N)|\lesssim_{\epsilon}N^{6-7\alpha+\epsilon},$

which our result improves. However, the estimates in [CS] extend to the case
of weighted sums and higher order phases, which ours does not.

In Section 4 we consider an analogue of a related problem raised by Chen and
Shparlinski. In [CS2] and [CS3] the authors analyze the following set which
characterizes points contributing to large values of $w_{N}$ as
$N\rightarrow\infty$:

$\mathcal{E}_{\alpha}=\\{(x,t)\in{\mathbb{T}}^{2}:|w_{N}(x,t)|\geq
N^{\alpha}\text{ for infinitely many }N\in\mathbb{N}\\}.$

It is not too hard to show that $|\mathcal{E}_{\alpha}|=0$ if $\alpha>1/2,$
since on ‘most’ of ${\mathbb{T}}^{2}$ we have square-root cancellation with
$|w_{N}|\lesssim N^{1/2}$ (see for example [CS2] for a proof). On the other
hand, understanding the Hausdorff dimension of $\mathcal{E}_{\alpha}$ is a
much more subtle problem.

Given a Borel set $\mathcal{S}$ we will write $\dim(\mathcal{S})$ to denote
its Hausdorff dimension, so that

$\dim(\mathcal{S})=\inf\\{s>0:\mathcal{H}^{s}(\mathcal{S})=0\\},\ \ \ \ \
\mathcal{H}^{s}=s\text{-dimensional Hausdorff measure}.$

In [CS2] and [CS3] Chen and Shparlinski proved that

(1.6)
$\min(3/2,3(1-\alpha))\leq\text{dim}(\mathcal{E}_{\alpha})\leq\frac{8(1-\alpha)}{2-\alpha}.$

It was also recently shown by Chen, Kerr, Maynard, and Shparlinski in [CKMS]
that there exist constants $a,b>0$ such that the set

$\\{(x,t)\in{\mathbb{T}}^{2}:aN^{1/2}\leq|w_{N}(x,t)|\leq bN^{1/2}\text{ for
infinitely many }N\in\mathbb{N}\\}$

has full Lebesgue measure (and in fact this holds for more general Weyl sums
associated to the moment curve).

We will consider a one-dimensional version of these types of estimates. Define

$\mathcal{O}_{\alpha}=\\{x\in{\mathbb{T}}:\sup_{0<t<1}|w_{N}(x,t)|\geq
N^{\alpha}\text{ for infinitely many }N\in\mathbb{N}\\}$

As a consequence of Theorem 1.1 we will obtain the following.

######  Proposition 1.1.

If $3/4\leq\alpha\leq 1$ then

$\dim(\mathcal{O}_{\alpha})=4(1-\alpha).$

We prove this result in Section 4. This proposition can be seen as a ‘Weyl sum
analogue’ of the divergence set estimates for the Euclidean Schrödinger
equation proved in [BBCR], [DZ], [LR]. Indeed the method of proof is similar.

We remark that the lower bound in the proof of Proposition 1.1 does not
require our maximal estimate. Instead it follows from the classical
observation that $\mathcal{O}_{\alpha}$ contains points which are well-
approximated by rationals with odd denominators. A theorem of Jarník gives us
good information about the dimension of such sets. The set
$\mathcal{O}_{\alpha}$ could contain many other points, but Theorem 1.1 will
be used to show that these additional points do not increase the Hausdorff
dimension.

###  1.2. Connection with recent estimates of Weyl sums

We briefly discuss some other recent estimates of Weyl sums that are related
to Theorem 1.1 and the corollaries mentioned above. There is a wide body of
research on Weyl sums, and we do not claim to give anything close to a
complete summary here.

Motivated by problems related to the Talbot effect from optics, the authors in
[BPPSV] have recently proved the following essentially sharp bound: for any
$\epsilon>0$ and $\gamma\in[0,1)$ one has

(1.7) $\sup_{x\in[0,1]}\big{|}\sum_{n=1}^{N}e^{2\pi
i(xn+(x+\gamma)n^{2})}\big{|}\lesssim_{\epsilon,\gamma}N^{3/4+\epsilon}.$

Erdoğan and Shakan had previously shown in [ErSh] that

(1.8) $\sup_{x\in[0,1]}\big{|}\sum_{n=1}^{N}e^{2\pi
i(xn+(a-rx)n^{2})}\big{|}\lesssim_{\epsilon,a,r}N^{4/5+\epsilon}.$

In [CS5] Chen and Shparlinski also prove analogues of these types of estimates
involving supremums over preimages of Lipschitz functions.

If more information is known about the coefficients one can obtain
improvements. For example, Oskolkov showed that if $r=m/2\pi$ and $a=k/2$ for
$m,k$ odd integers then the bound in (1.8) can be improved to $N^{1/2}$ (see
[Os], Lemma 2). For more information about the motivation for these estimates
and their relation to the Talbot effect see for example [ErT], Section 2.3, or
the introductions of [ErSh] and [Os].

Note that the dependence on $N$ in (1.7) is the same as in our $L^{4}$ maximal
estimate. There are some overlapping ideas in parts of the proof of Theorem
1.1 and the proof of (1.7), in particular related to the role of Diophantine
approximation of the coefficient of $n^{2}$. However, neither of these results
appears to imply the other. Indeed, to prove Theorem 1.1 one must understand
the size of the region where the supremum in $t$ is much larger than
$N^{3/4}$. Also note that the constant in (1.7) depends on $\gamma$, which
leads to additional obstacles (for example if we shift the frequencies and try
to exploit Galilean invariance). On the other hand, Theorem 1.1 implies that
if $t(x)$ is any measurable function of $x$ (and in particular a linear
function of $x$) then

$\big{(}\int_{{\mathbb{T}}}\big{|}\sum_{n=1}^{N}e^{2\pi
i(xn+t(x)n^{2})}\big{|}^{4}dx\big{)}^{1/4}\leq C_{\epsilon}N^{3/4+\epsilon},$

but this is not enough to conclude (1.7).

The inequalities (1.7), (1.8) imply dimension bounds on the graphs of certain
solutions to the linear Schrödinger equation if they are restricted to lines,
as studied in [ErSh], [Os], and [ChOs]. They are also useful for proving
similar estimates for the Hilbert-transform-like function

$H(x,t)=\sum_{\begin{subarray}{c}n\in{\mathbb{Z}}\\\ n\neq
0\end{subarray}}\frac{1}{n}e^{2\pi i(xn+tn^{2})},$

which is studied in depth by Chakhkiev and Oskolkov in [ChOs]. Indeed, by
using Littlewood-Paley theory and summation by parts one can use estimates for
the Weyl sum to obtain information about $H$; estimates for $H$ then imply
bounds on the dimension of graphs of solutions to the linear Schrödinger
equation with initial data of bounded variation (see for example [ErSh],
Theorem 2.2.). In [ChOs] the authors consider $H$ restricted to slices with
$x$ or $t$ fixed, and characterize the Hölder continuity of these slices in
terms of the Diophantine approximation properties of $x$ or $t$. In
particular, in [ChOs], Corollary 4, the authors calculate the Hausdorff
dimension of the set of $x$ for which the restriction $H|_{x}$ is
$\alpha$-Hölder continuous, and prove analogous bounds for $H|_{t}$. The
methods of proof bear a resemblance to the proof of the lower bound in
Proposition 1.1, and in particular the Jarník theorem (Theorem 4.1 below)
plays an important role in their argument.

###  1.3. Notation

We will write $A\lesssim B$ to mean there is a uniform constant $C>0$ such
that $A\leq CB$. If there is a constant $C(\beta)$ that depends on some
parameter $\beta$ such that $A\leq C(\beta)B$ we will write
$A\lesssim_{\beta}B$. We similarly define $A\gtrsim B$ and
$A\gtrsim_{\beta}B,$ and write $A\sim B$ if $A\lesssim B$ and $A\gtrsim B$.
Throughout the paper we occasionally let $C,c,c^{\prime},$ etc. denote uniform
constants which may change from line-to-line. We also let $C_{\epsilon}$ be
the distinguished constant appearing in Theorem 1.1. Once we fix $\epsilon>0$
the constant $C_{\epsilon}$ does not change line-to-line.

### Acknowledgments

The author thanks the anonymous referees for providing helpful comments which
have improved the presentation of the paper. The author also thanks M. B.
Erdoğan for suggesting several references and for helpful discussions related
to the material in this paper.

##  2\. The proof of Theorem 1.1

Certain collections of axis-parallel rectangles will play a key role in the
proof of Theorem 1.1. Fix $3/4\leq\alpha\leq 1$ and a small $\eta>0$, and tile
$[0,1]^{2}$ by axis-parallel rectangles of dimensions approximately
$N^{-2+\alpha-\eta}\times N^{-4+2\alpha-2\eta}$. Let $\Lambda_{\alpha}$ denote
the rectangles in this tiling. We will work with certain sub-collections of
$\Lambda_{\alpha}$.

######  Definition 2.1.

Let $\pi_{x}:[0,1]^{2}\rightarrow[0,1]$ denote projection in the $x$ variable.
Let $\mathcal{Q}$ be a sub-collection of $\Lambda_{\alpha}$. We say that
$\mathcal{Q}$ is one-dimensional at scale $(N,\alpha)$ if for each
$x_{0}\in[0,1]$ there are at most two rectangles $Q\in\mathcal{Q}$ such that
$x_{0}\in\pi_{x}(Q)$

The choice of scale is motivated by the following ‘locally-constant’ behavior
of exponential sums.

######  Lemma 2.1.

Suppose $0<\alpha\leq 1$ and fix a small $\eta>0$. Suppose
$cN^{\alpha}\leq|w_{N}(x_{0},t_{0})|\leq CN^{\alpha}$ and let $Q_{\alpha}$ be
an axis-parallel rectangle of dimensions $N^{-2+\alpha-\eta}\times
N^{-3+\alpha-\eta}$ that contains $(x_{0},t_{0})$. Then
$N^{\alpha}\lesssim|w_{N}(x,t)|\lesssim N^{\alpha}$ for all $(x,t)\in
Q_{\alpha}$.

###### Proof.

This is a direct consequence of the mean value theorem (see also [Wo2] or [CS]
for more general versions of this estimate).∎

Below we suppress the role of $\eta$ from the notation, since it can be chosen
to be much smaller than all other small parameters under consideration (e.g.
take $\eta=\epsilon^{100}$).

We now turn to the proof of Theorem 1.1. To slightly simplify the notation we
will write

$u(x,t)=w_{N}(x,t).$

Fix $\epsilon>0$ for the rest of the argument. If $\sup_{t}|u(x,t)|\leq
N^{3/4}$ then the estimate is trivial, so it suffices to estimate the integral
on the set $E$ where the supremum is larger than $N^{3/4}$. Partition $E$ into
a disjoint union of sets $E_{k}$ where the supremum is in $[2^{k},2^{k+1}),$
with $N^{3/4}\leq 2^{k}\leq N.$ The number of $k$ is $O(\log N)$ and so it
suffices to estimate

(2.1) $\big{(}\int_{E_{k}}\sup_{0<t<1}|u(x,t)|^{4}dx\big{)}^{1/4}$

Our argument will involve induction on the scale $N$. In particular note that
since Theorem 1.1 is trivial if $N\lesssim_{\epsilon}1$ we may assume by
induction that it holds for any scale $N^{\prime}$ with $N^{\prime}<N/2$.

Choose $\alpha\in[3/4,1]$ such that $2^{k}\sim N^{\alpha}$, and cover $E_{k}$
by a union of intervals $I_{\alpha}$ of length approximately $N^{-2+\alpha}$
which overlap only at their boundaries. For each $I_{\alpha}$ pick a point
$x_{\alpha}\in I_{\alpha}\cap E_{k}$ and choose an interval $J_{\alpha}$ of
length approximately $N^{-4+2\alpha}$ such that $\sup_{t}|u(x_{\alpha},t)|$ is
attained at some $t_{\alpha}\in J_{\alpha}$. Let

$Q_{\alpha}=I_{\alpha}\times J_{\alpha}.$

Then the $Q_{\alpha}$ form a one-dimensional collection at scale $(N,\alpha)$.
Moreover, by Lemma 2.1 there exist constants $c,C>0$ such that

$cN^{\alpha}\leq|u(x,t)|\leq CN^{\alpha},\ \ \ (x,t)\in Q_{\alpha},$

and so

(2.2)
$\big{(}\int_{E_{k}}\sup_{0<t<1}|u(x,t)|^{4}dx\big{)}^{1/4}N^{1-\alpha/2}\big{(}\sum_{Q_{\alpha}}\int_{Q_{\alpha}}|u(x,t)|^{4}dxdt\big{)}^{1/4}.$

Let $X$ denote the one-dimensional collection of rectangles $Q_{\alpha}$
appearing in $\eqref{eq:oneDim}$.

###  2.1. Partitioning the rectangles in $X$

We will now partition the rectangles in $X$ into different groups $X_{q}$
according to the rational approximation properties of $t$ with $(x,t)\in
Q_{\alpha}$. To clear up notation we now drop the subscript $\alpha$ from the
$Q_{\alpha}\in X$.

We will use the following dispersive estimate for $u$ due to Bourgain, which
is a refinement of the classical Weyl bound.

######  Lemma 2.2 ([B]).

Suppose there are integers $1\leq q\leq N$ and $0\leq a<q$ with $(a,q)=1$ such
that $|t-a/q|\leq 1/qN.$ Then for all $x$ one has

$|u(x,t)|\lesssim\log N\frac{N}{q^{1/2}(1+N|t-a/q|^{1/2})}.$

###### Proof.

This is essentially proved in Lemma 3.18 in [B], although the details are only
provided for sums of the form

$W(x,t)=\sum_{|n|\lesssim N}\phi(n/N)e^{2\pi i(nx+n^{2}t)},$

where $\phi$ is a smooth cut-off equal to 1 on $[1,N]$ (for example). But we
can write $u=W(\cdot,t)\ast D_{N}(x),$ where $D_{N}$ is the Dirichlet kernel.
Therefore

$|u(x,t)|\leq\|W(\cdot,t)\|_{L^{\infty}}\|D_{N}\|_{L^{1}}\lesssim\log(N)\|W(\cdot,t)\|_{L^{\infty}},$

using the well-known bound $\|D_{N}\|_{L^{1}}\lesssim\log(N)$. The claim now
follows by applying Lemma 3.18 from [B]. ∎

We choose $m\in\mathbb{N}$ such that $2^{-m/2}N\sim N^{\alpha}$, so $2^{m}$ is
the dyadic scale of $N^{2-2\alpha}$. This choice of scale is motivated by
Lemma 2.2. Note that $2^{m}\lesssim N^{1/2}$ since $\alpha\geq 3/4$.

Let $Q_{j}=I_{j}\times J_{j}$ index $Q\in X$. Then by Dirichlet’s
approximation theorem, for each $t\in J_{j}$ there exist integers $0\leq
a<q\leq N$ with $|t-a/q|<1/Nq$ and $(a,q)=1$. Lemma 2.2 then implies that
there must exist some $t_{j}\in J_{j}$ and some integers $0\leq
a_{j}<q_{j}\leq N$ with $(a_{j},q_{j})=1$ such that

$2^{-m/2}N\lesssim\|u\|_{L^{\infty}(Q_{j})}\lesssim\frac{N\log(N)}{q_{j}^{1/2}(1+N|t_{j}-a_{j}/q_{j}|^{1/2})}.$

In particular

(2.3) $1\leq q_{j}\leq c\log(N)^{2}2^{m}$

and

(2.4) $|t_{j}-a_{j}/q_{j}|\leq c\log(N)^{2}\frac{2^{m}}{q_{j}N^{2}}.$

Note that (2.4) guarantees $t_{j}$ is closer to the rational $a_{j}/q_{j}$
than what Dirichlet’s theorem implies. We collect this result as the following
lemma.

######  Lemma 2.3.

Suppose $Q$ is an axis-parallel rectangle of dimension $N^{-2+\alpha}\times
N^{-4+2\alpha}$ with $\|u\|_{L^{\infty}(Q)}\geq N^{\alpha}$, where $u$ is a
scale-$N$ quadratic Weyl sum. Suppose $t_{\ast}$ is the time at which the
supremum is attained, and $|t_{\ast}-a/q|\leq\frac{1}{qN}$ with $0\leq a<q\leq
N$ and $(a,q)=1$. Then if $2^{m}\sim N^{2-2\alpha}$ we have

(2.5) $1\leq q\leq c\log(N)^{2}2^{m}$

and in fact

(2.6) $|t_{\ast}-a/q|\leq c\log(N)^{2}\frac{2^{m}}{qN^{2}}.$

We now group $X$ according to the value of $q_{j}$ from the range (2.3). We
say $q_{j}\sim Q_{j}$ if

$|t_{j}-a_{j}/q_{j}|\leq c\log(N)^{2}\frac{2^{m}}{q_{j}N^{2}}$

for some $0\leq a_{j}<q_{j}$ with $(a_{j},q_{j})=1$ and

$1\leq q_{j}\leq c\log(N)^{2}2^{m},$

where $c$ is the constant appearing above. Note that $a_{j}/q_{j}$ is uniquely
determined. Indeed, if $a^{\prime}/q^{\prime}\neq a_{j}/q_{j}$ and both
$q_{j}\sim Q_{j}$ and $q^{\prime}\sim Q_{j}$ then we must have

$\frac{1}{q^{\prime}q_{j}}\leq\big{|}\frac{a^{\prime}}{q^{\prime}}-\frac{a_{j}}{q_{j}}\big{|}\leq
2c\log(N)^{2}\frac{2^{m}}{N^{2}}\big{(}\frac{1}{q^{\prime}}+\frac{1}{q_{j}}\big{)}$

and therefore $N^{2}\lesssim\log(N)^{2}2^{2m}$; but this is false since
$2^{2m}\lesssim N$. A similar argument shows that if $q^{\prime}=q_{j}$ then
$a^{\prime}=a_{j}$ whenever $q_{j}\sim Q_{j}$.

We define

$X_{q}=\\{Q_{j}\in X:q\sim Q_{j}\\}.$

Then

$X=\bigcup_{q=1}^{c\log(N)^{2}2^{m}}X_{q}$

and the union is disjoint.

###  2.2. Contribution of $X_{q}$ with $1\leq q\leq N^{\delta}$

We begin by considering the contribution to (2.2) from rectangles in $X_{q}$
with $1\leq q\leq N^{\delta}$, where $\delta>0$ is a small parameter with
$\delta<\epsilon^{2}$. In this case we directly estimate the integral in (2.2)
without using induction. Note that for each $Q_{j}$ under consideration we
have

$|t_{j}-a/q|\leq c\log(N)^{2}\frac{2^{m}}{qN^{2}}$

for some choice of $1\leq q\leq N^{\delta}$ and $0\leq a<q$ with $(a,q)=1$.
Let $\Phi(q)$ denote the number of positive integers $a$ less than $q$ such
that $(a,q)=1$ (so $\Phi$ is the Euler totient function). Then the number of
possible $a/q$ in this case is 111 In fact we have the more precise estimate
$\sum_{q=1}^{N^{\delta}}\Phi(q)=\frac{3}{\pi^{2}}N^{2\delta}+O(\delta
N^{\delta}\log(N))$ but this will be of no use here. See [A], Theorem 3.7, for
a proof of this identity.

(2.7) $\sum_{q=1}^{N^{\delta}}\Phi(q)\lesssim N^{2\delta}.$

Therefore after a loss of $CN^{2\delta}$ we may assume that there is one fixed
$a/q$ as above such that all remaining $Q_{j}$ satisfy $|t_{j}-a/q|\leq
c\log(N)^{2}\frac{2^{m}}{qN^{2}}$. Let $Y$ denote the collection of the
remaining $Q$. Then by (2.3) we have

$\bigcup_{Q\in Y}Q\subset[0,1]\times[a/q-c\log(N)^{2}\frac{2^{m}}{qN^{2}},\
a/q+c\log(N)^{2}\frac{2^{m}}{qN^{2}}]:={\mathbb{T}}\times I,$

with $|I|\lesssim_{\delta}2^{m}N^{-2+\delta}\sim N^{-2\alpha+\delta}.$ Since
$\delta<\epsilon^{2}$ all losses of $N^{O(\delta)}$ will be negligible. We
will suppress the dependence below and write $A\lessapprox B$ to mean $A\leq
C(\delta)N^{O(\delta)}B$.

Notice that since the collection $Y$ is one-dimensional and contained in
${\mathbb{T}}\times I$ we have

(2.8) $N^{1-\alpha/2}\big{(}\sum_{Q\in
Y}\int_{Q}|u|^{4}\big{)}^{1/4}\lesssim\big{(}\int_{{\mathbb{T}}}\sup_{t\in
I}|u(x,t)|^{4}dx\big{)}^{1/4}.$

To conclude the argument we will appeal to the following local estimate of
Moyua and Vega.

######  Theorem 2.1 ([MV]).

Let $u$ be any solution to the linear Schrödinger equation on ${\mathbb{T}}$
(not necessarily a Weyl sum). Suppose $u(x,0)=f(x)$ and $\widehat{f}$ is
supported in $[-N,N]$. If $\eta>0$ then for any $\epsilon>0$ we have

$\|\sup_{0<t<\eta}|u(x,t)|\|_{L^{4}({\mathbb{T}})}\lesssim_{\epsilon}N^{1/2+\epsilon}\max(N^{-1},\eta)^{1/4}\|f\|_{L^{2}({\mathbb{T}})}.$

When $\eta\sim N^{-1}$ this implies the sharp bound for the maximal operator
with a restricted supremum. Note that in the ‘Euclidean window’ $0<t<N^{-2}$
we expect matters to reduce to known estimates for the solution on
${\mathbb{R}}$; this result gives an extension to the interval $0<t<N^{-1}$.
The theorem applies to any time interval of length $\eta$ since the interval
can be translated by modulating the Fourier coefficients of $f$, and this does
not change $\|f\|_{L^{2}}$.

Applying Theorem 2.1 yields

(2.9) $\big{(}\int_{{\mathbb{T}}}\sup_{t\in
I}|u(x,t)|^{4}dx\big{)}^{1/4}\lessapprox N^{3/4}$

(recall $|I|\lessapprox N^{-2\alpha}<N^{-1}$). Then from (2.8) and (2.9) we
conclude that

$N^{1-\alpha/2}\big{(}\sum_{Q\in
Y}\int_{Q}|u(x,t)|^{4}dxdt\big{)}^{1/4}\lessapprox N^{3/4},$

as desired.

###  2.3. Contribution of $X_{q}$ with $q>N^{\delta}$

We now turn to the rectangles belonging to some $X_{q}$ with $q>N^{\delta}$.
We will control the contribution of each $X_{q}$ by using induction on the
scale $N$.

Suppose $(x,t)\in Q_{j}$ and $Q_{j}\in X_{q}$. We break the range of summation
in $u$ into congruence classes mod $q$ to write

(2.10) $\displaystyle u(x,t)$ $\displaystyle=\sum_{r=1}^{q}\sum_{l=0}^{\lfloor
N/q\rfloor}e^{2\pi i((lq+r)x+(lq+r)^{2}t)}+O(q)$
$\displaystyle=\sum_{r=1}^{q}e^{2\pi i(r^{2}t+rx)}\sum_{l=0}^{\lfloor
N/q\rfloor}e^{2\pi i(l^{2}q^{2}t+lqx)}e^{2\pi i(2rlqt)}+O(q).$

By construction there is some integer $a$ with $(a,q)=1$ such that

$|t_{j}-a/q|\lesssim\log(N)^{2}\frac{2^{m}}{qN^{2}}.$

Since $|t-t_{j}|\leq 1/N^{2}$ and $q\lesssim\log(N)^{2}2^{m}$ we also have
$|t-a/q|\lesssim\log(N)^{2}\frac{2^{m}}{qN^{2}}.$ The choice of $a$ depends on
$Q_{j}$, but below we will use a pointwise estimate that is uniform over
possible values of $a$.

Since $|t-a/q|\lesssim\log(N)^{2}\frac{2^{m}}{qN^{2}}$ we have

$2rlq|t-a/q|\lesssim\log(N)^{2}2^{m}N^{-1}$

and therefore

(2.11) $e^{2\pi i(2rlqt)}=e^{2\pi i(2rlq[t-a/q])}=1+O(\log(N)^{2}2^{m}N^{-1})$

with the implicit constant independent of $a$ and $q$. It follows from (2.3),
(2.10), and (2.11) that we can write

(2.12) $u(x,t)=\sum_{r=1}^{q}e^{2\pi i(r^{2}t+rx)}\sum_{l=0}^{\lfloor
N/q\rfloor}e^{2\pi i(l^{2}q^{2}t+lqx)}+E_{q}(x,t),\ \ \ \ \ (x,t)\in Q\in
X_{q}$

where $E_{q}$ is supported on $\bigcup_{Q\in X_{q}}Q$ and

$|E_{q}(x,t)|\mathbbm{1}_{Q}(x,t)\lesssim
q\frac{N}{q}\log(N)^{2}2^{m}N^{-1}\lesssim\log(N)^{2}2^{m},\ \ \ \ \ Q\in
X_{q}$

(note that we have absorbed the error $O(q)$ from (2.10) into the definition
of $E_{q}$). Define the function $E(x,t)=\sum_{q}E_{q}(x,t)$. Since the cubes
$Q\in X$ are pairwise disjoint we have

$|E(x,t)|\lesssim\log(N)^{2}2^{m},\ \ \ \ (x,t)\in\bigcup_{Q\in X}Q.$

Also define functions $v_{q}(x,t)$ and $w_{q}(x,t)$ by setting

$\displaystyle v_{q}(x,t)=\sum_{r=1}^{q}e^{2\pi
i(r^{2}t+rx)}\sum_{l=0}^{\lfloor N/q\rfloor}e^{2\pi i(l^{2}q^{2}t+lqx)},$
$\displaystyle w_{q}(x,t)=\sum_{l=0}^{\lfloor N/q\rfloor}e^{2\pi
i(l^{2}q^{2}t+lqx)},\ \ \ \ (x,t)\in\bigcup_{Q\in X_{q}}Q.$

We will now argue by induction on the scale. The main idea is that each
$w_{q}$ can be viewed as a Weyl sum at scale $\sim N/q$, and so we can hope to
use our inductive hypothesis to count the number of cubes in $X_{q}$.

Note that since $|t-a/q|<1/q^{2}$ for some $(a,q)=1$ the classical Weyl bound
tells us that

$|v_{q}(x,t)|\lesssim(\log(q)q)^{1/2}|w_{q}(x,t)|.$

It now follows from the above estimates that

(2.13) $\displaystyle\big{(}\sum_{Q\in
X_{q}}\int_{Q}|u(x,t)|^{4}dxdt\big{)}^{1/4}\lesssim$
$\displaystyle\big{(}\log(q)^{2}q^{2}\sum_{Q\in
X_{q}}\int_{Q}|w_{q}(x,t)|^{4}dxdt\big{)}^{1/4}$
$\displaystyle+\big{(}\int_{\bigcup_{Q\in
X_{q}}}|E(x,t)|^{4}dxdt\big{)}^{1/4}$

and hence

(2.14) $\displaystyle\sum_{q=N^{\delta}}^{c2^{m}\log(N)^{2}}\sum_{Q\in
X_{q}}\int_{Q}|u(x,t)|^{4}dxdt\lesssim\sum_{q=N^{\delta}}^{c2^{m}\log(N)^{2}}$
$\displaystyle\log(q)^{2}q^{2}\sum_{Q\in X_{q}}\int_{Q}|w_{q}(x,t)|^{4}dxdt$
$\displaystyle+O(N^{-4+2\alpha}2^{4m}\log(N)^{8}),$

with the second term on the right coming from the bound for the error $|E|$.
Since

$2^{m}\log(N)^{2}<N^{3/4}$

the error term makes an acceptable contribution to (2.2) and we henceforth
drop it from our discussion.

Now observe that if we set

$w(x,t)=\sum_{l=0}^{\lfloor N/q\rfloor}e^{2\pi i(lx+l^{2}t)}$

then we have

(2.15) $w_{q}(x,t)=w(qx,q^{2}t).$

This identity puts us in a position to use induction at scale $N/q$, although
we need to do a little more work to ensure that the induction hypotheses are
satisfied.

It is evident that $w_{q}$ is $(1/q)$-periodic in the $x$ variable and
$(1/q^{2})$-periodic in the $t$ variable. We exploit this fact with the
following lemma.

######  Lemma 2.4.

Fix $q$ and define the collection $X_{q}$ as above. Then there exists a
collection of rectangles $\overline{X}_{q}$ such that

  1. (1)

Each $\overline{Q}\in\overline{X}_{q}$ has dimensions approximately
$N^{-2+\alpha}\times N^{-4+2\alpha}$

  2. (2)

$\overline{Q}\subset[0,1/q]\times[0,1/q^{2}]$ for each
$\overline{Q}\in\overline{X}_{q}$

  3. (3)

We have

$\sum_{Q\in
X_{q}}\int_{Q}|w_{q}(x,t)|^{4}dxdt\lesssim\sum_{\overline{Q}\in\overline{X}_{q}}\int_{\overline{Q}}|w_{q}(x,t)|^{4}dxdt.$

  4. (4)

The collection $\overline{X}_{q}$ is a union of at most $O(q)$ one-dimensional
collections of rectangles.

###### Proof.

The first three points follow from the fact that $w_{q}(x,t)=w(qx,q^{2}t)$ as
described above, hence $w_{q}(x,t)$ is $(1/q)$-periodic in $x$ and
$(1/q^{2})$-periodic in $t$. Indeed we can let $\overline{Q}$ be the translate
of $Q$ in $[0,1/q]\times[0,1/q^{2}]$ such that

$w_{q}(x+1/q,t+1/q^{2})\mathbbm{1}_{Q}(x+n/q,t+m/q^{2})=w_{q}(x,t)\mathbbm{1}_{\overline{Q}}(x,t),\
\ \ \ (x,t)\in[0,1/q]\times[0,1/q^{2}]$

for some integers $n,m$. The last point (4) follows from the fact that
$\overline{Q}$ constructed in this matter can only overlap on the order of
$O(q)$, since $X_{q}$ is one-dimensional. ∎

It is here that we are crucially using the one-dimensional hypothesis on the
collection of rectangles, since this property ensures that there is limited
overlap when shifting the rectangles as in Lemma 2.4. We will see below that
overlap on the order of $O(q)$ is admissible.

To make the last point more precise, define a map $Q\rightarrow\tau(Q)$ by
letting $\tau(Q)=\overline{Q}$ be the copy of $Q$ mod $(1/q,1/q^{2})$ chosen
in Lemma 2.4. Recall from (2.13) that up to negligible error terms we have

$\sum_{Q\in X_{q}}\int_{Q}|u(x,t)|^{4}dxdt\lesssim\log(q)^{2}q^{2}\sum_{Q\in
X_{q}}\int_{Q}|w_{q}(x,t)|^{4}dxdt.$

By Lemma 2.4 we may therefore find a sub-collection $X^{\prime}_{q}\subset
X_{q}$ such that $\tau(X^{\prime}_{q})$ is a one-dimensional collection
contained in $[0,1/q]\times[0,1/q^{2}]$ and such that

(2.16) $\sum_{Q\in
X_{q}}\int_{Q}|u(x,t)|^{4}dxdt\lesssim\log(q)^{2}q^{3}\sum_{Q\in
X_{q}^{\prime}}\int_{\tau(Q)}|w_{q}(x,t)|^{4}dxdt.$

Since the $\tau(Q)$ are contained in $[0,1/q]\times[0,1/q^{2}]$ we can make a
change of variables $(y,s)=(qx,q^{2}t)$ to obtain

(2.17) $\log(q)^{2}q^{3}\sum_{Q\in
X_{q}^{\prime}}\int_{\tau(Q)}|w_{q}(x,t)|^{4}dxdt\leq\log(q)^{2}\sum_{R\in
Y_{q}}\int_{R}|w(x,t)|^{4}dxdt,$

where $Y_{q}$ is the image of $X_{q}^{\prime}$ under the rescaling. In
particular each $R\in Y_{q}$ is an axis-parallel rectangle of dimensions
approximately

$qN^{-2+\alpha}\times q^{2}N^{-4+2\alpha},$

and each $R\in Y_{q}$ is contained in $[0,1]^{2}$. Moreover $Y_{q}$ is one-
dimensional. Note that we have applied a version of the parabolic rescaling
that plays a crucial role in the theory of the Euclidean operator
$e^{it\Delta_{{\mathbb{R}}^{n}}}$.

Now using (2.16), (2.17), and the fact that $Y_{q}$ is one-dimensional, we
conclude that

(2.18) $\sum_{Q\in X_{q}}\int_{Q}|u(x,t)|^{4}dxdt\lesssim
N^{-4+2\alpha}\log(q)^{2}q^{2}\int_{{\mathbb{T}}}\sup_{0<t<1}|w(x,t)|^{4}dxdt.$

Recall that $w(x,t)=\sum_{n=1}^{\lfloor N/q\rfloor}e^{2\pi i(nx+n^{2}t)},$ so
$w$ is a scale $N/q$ Weyl sum. Then by our induction hypothesis

$\int_{{\mathbb{T}}}\sup_{0<t<1}|w(x,t)|^{4}dx\leq
C_{\epsilon}^{4}N^{4\epsilon}q^{-4\epsilon}N^{3}q^{-3}.$

Inserting this into (2.18) yields

(2.19) $N^{4-2\alpha}\sum_{Q\in X_{q}}\int_{Q}|u(x,t)|^{4}dxdt\leq
CC_{\epsilon}^{4}q^{-4\epsilon}\log(q)^{2}q^{-1}N^{3+4\epsilon}$

Recall $q\geq N^{\delta}$ and $\delta=\delta(\epsilon)$. We may therefore
assume $N\gtrsim_{\epsilon}1$ has been chosen large enough that

$Cq^{-2\epsilon}\log(q)^{2}\leq 1.$

Then by plugging (2.19) into (2.14) we arrive at

$\displaystyle N^{4-2\alpha}\sum_{q=N^{\delta}}^{c2^{m}\log(N)^{2}}\sum_{Q\in
X_{q}}\int_{Q}|u(x,t)|^{4}dxdt$ $\displaystyle\leq
N^{-2\delta\epsilon}\sum_{q=N^{\delta}}^{c\log(N)^{2}2^{m}}CC_{\epsilon}^{4}q^{-1}N^{3+4\epsilon}$
$\displaystyle\leq
N^{-2\delta\epsilon}C\log(N)^{3}(C_{\epsilon}N^{3/4+\epsilon})^{4}.$

Finally we may assume that $N\gtrsim_{\epsilon}1$ has been chosen large enough
that

$N^{-\delta\epsilon}C\log(N)^{3}\leq(1/3)^{4}.$

This shows that rectangles from $X_{q}$ with $q\geq N^{\delta}$ give an
acceptable contribution to (2.2).

This was the only remaining case, and so we conclude from (2.2) that

$\displaystyle\big{(}\int_{{\mathbb{T}}}\sup_{0<t<1}|u(x,t)|^{4}dx\big{)}^{1/4}$
$\displaystyle\lesssim\log(N)^{1/4}\big{(}\int_{E_{k}}\sup_{0<t<1}|u(x,t)|^{4}dx\big{)}^{1/4}$
$\displaystyle\lesssim\log(N)^{1/4}N^{1-\alpha/2}\big{(}\sum_{Q_{\alpha}}\int_{Q_{\alpha}}|u(x,t)|^{4}dxdt\big{)}^{1/4}$
$\displaystyle\leq
C\log(N)^{1/4}N^{-\delta\epsilon}C_{\epsilon}N^{3/4+\epsilon}$
$\displaystyle\leq C_{\epsilon}N^{3/4+\epsilon},$

the last line following as long as we choose $\delta<\epsilon^{2}$ such that
$CN^{-\delta\epsilon}\log(N)^{1/4}\leq 1$. This completes the proof of Theorem
1.1.

######  Remark 2.1.

In this section we have worked with an equivalent version of the maximal
problem that involves estimating $w_{N}$ on one-dimensional collections of
axis-parallel rectangles in $\mathbb{T}^{2}$. We have chosen to present the
argument in this manner to illustrate the connection with ideas of Du and
Zhang [DZ], who prove the Euclidean analogue of Theorem 1.1 by analyzing the
behavior of the associated Fourier extension operator on lower-dimensional
unions of cubes in space-time. It is straightforward (and perhaps more
natural) to rewrite the argument in this section to directly involve the
maximal function. Indeed, the rectangles in $X_{q}$ can be replaced by
intervals consisting of their projections to $[0,1]$, and the two-dimensional
integrals used in this section can be replaced by one-dimensional integrals of
the maximal function. The argument then goes through as above, with minor
changes made.

##  3\. Corollaries of the maximal estimate

As a corollary of Theorem 1.1 we obtain the following discrete ‘one-
dimensional’ level set estimate for $w_{N}$.

######  Proposition 3.1.

Let $w_{N}$ denote a scale-$N$ quadratic Weyl sum as above. Let $A>0$ be an
arbitrary constant independent of other parameters. Fix $3/4\leq\alpha<1$. Let
$X$ be a one-dimensional collection of rectangles $Q$ at scale $(N,\alpha)$
and suppose

$\|w_{N}\|_{L^{\infty}(Q)}\geq AN^{\alpha}$

for each $Q$. Then for every $\epsilon>0$ there exists $C(\epsilon,A)>0$ such
that

$\\#X\leq C(\epsilon,A)N^{\epsilon}N^{5(1-\alpha)}.$

Notice that the number of rectangles in a one-dimensional collection at scale
$(N,\alpha)$ is always less than $CN^{2-\alpha}$, and so the proposition is
trivial in the range $\alpha\leq 3/4$.

###### Proof.

This is a consequence of the maximal estimate along with Lemma 2.1. Indeed we
have

$\big{(}\sum_{Q\in X}\int_{Q}|w_{N}(x,t)|^{4}dxdt\big{)}^{1/4}\gtrsim
N^{\alpha}N^{-6/4+3\alpha/4}(\\#X)^{1/4},$

and on the other hand since the collection is one-dimensional

$\big{(}\sum_{Q\in X}\int_{Q}|w_{N}(x,t)|^{4}dxdt\big{)}^{1/4}\lesssim
N^{-1+\alpha/2}\big{(}\int_{{\mathbb{T}}}\sup_{0<t<1}|w_{N}(x,t)|^{4}dx\big{)}^{1/4}\leq
C_{\epsilon}N^{\epsilon}N^{-1/4+\alpha/2}.$

Therefore

$(\\#X)^{1/4}\lesssim C_{\epsilon}N^{2\epsilon}N^{5/4-5\alpha/4}$

which implies the proposition. ∎

Proposition 3.1 implies the following.

######  Corollary 3.1.

Let $\mathcal{S}_{\alpha}(N)$ denote the set of $x\in{\mathbb{T}}$ such that

$\sup_{0<t<1}\bigg{|}\sum_{n=1}^{N}e^{2\pi i(xn+tn^{2})}\bigg{|}\geq
cN^{\alpha}.$

Then $\mathcal{S}_{\alpha}(N)$ has measure
$O_{\epsilon}(N^{3-4\alpha+\epsilon}).$

The corollary follows by choosing intervals of the form
$[a/N^{-2+\alpha},(a+1)/N^{-2+\alpha}]$ which contain a point from
$S_{\alpha}$ and then applying Proposition 3.1.

As a consequence of Theorem 1.1 we also obtain the following refined
Strichartz-type estimate, which should be compared to the main estimates of Du
and Zhang in [DZ].

######  Corollary 3.2.

Let $w_{N}$ be a scale $N$ quadratic Weyl sum. Let $Q_{j}$ be a collection
one-dimensional rectangles in $[0,1]^{2}$ at scale $(N,1)$ (so the dimensions
are approximately $N^{-1}\times N^{-2}$). Then for any $\epsilon>0$ one has

$\big{(}\int_{\bigcup_{j}Q_{j}}|w_{N}(x,t)|^{4}dxdt\big{)}^{1/4}\lesssim_{\epsilon}N^{1/4+\epsilon}.$

###### Proof.

This is an immediate consequence of Theorem 1.1 since the collection is one-
dimensional. ∎

Corollary 3.2 is sharp, as is seen by taking one of the $Q_{j}$ to contain
$[0,10^{-6}N^{-1}]\times[0,10^{-6}N^{-2}]$.

Finally, the following is also an easy consequence of Theorem 1.1.

######  Corollary 3.3.

Let $q$ be an integer with $1\leq q<N$. Suppose
$\mathcal{P}=\\{q,2q,3q,...,kq\\}$ is contained in $[1,N]$, so that
$\\#\mathcal{P}\leq\lfloor N/q\rfloor$. Then for every $\epsilon>0$ there is
$C_{\epsilon}>0$ such that

$\bigg{\|}\sup_{0<t<1}\big{|}\sum_{n\in\mathcal{P}}e^{2\pi
i(n(\cdot)+n^{2}t)}\big{|}\bigg{\|}_{L^{4}({\mathbb{T}})}\leq
C_{\epsilon}q^{-3/4}N^{3/4+\epsilon}.$

###### Proof.

We have

$u(x,t)=\sum_{l=1}^{\lfloor N/q\rfloor}e^{2\pi
i(lqx+l^{2}q^{2}t)}=w(qx,q^{2}t),$

where

$w(x,t)=\sum_{l=1}^{\lfloor N/q\rfloor}e^{2\pi i(lx+l^{2}t)}.$

In particular $u(x,t)$ is $(1/q)$-periodic in $x$ and $(1/q^{2})$-periodic in
$t$. This implies that

$\int_{0}^{1}\sup_{0<t<1}|u(x,t)|^{4}dx=q\int_{0}^{1/q}\sup_{0<t<q^{-2}}|w(qx,q^{2}t)|^{4}dx.$

Making the change of variables $(y,s)=(qx,q^{2}t)$ then yields

$\int_{0}^{1}\sup_{0<t<1}|u(x,t)|^{4}dx=\int_{0}^{1}\sup_{0<s<1}|w(y,s)|^{4}dy.$

The claimed estimate now follows from Theorem 1.1. ∎

Note that if $u(x,t)$ is as in the previous corollary then
$\|u(\cdot,0)\|_{L^{2}}\sim(N/q)^{1/2}$. So the corollary says

$\|\sup_{0<t<1}|u(\cdot,t)\|_{L^{4}({\mathbb{T}})}\leq
C_{\epsilon}N^{\epsilon}q^{-1/4}N^{1/4+\epsilon}\|u(\cdot,0)\|_{L^{2}({\mathbb{T}})},$

which is an improvement over what can be expected in general (since the loss
of $N^{1/4}$ is sharp for the complete Weyl sum, for example).

##  4\. The dimension of $\mathcal{O}_{\alpha}$

We now prove Proposition 1.1. To prove the upper bound in the proposition we
will apply our maximal estimate to a modification of the Weyl sum, following
the completion method used by Chen and Shparlinski (see for example [CS3],
Lemma 2.2). We will also use a fractal version of Theorem 1.1 which can be
obtained by an argument of Eceizabarrena and Lucà [EL]. To prove the lower
bound we will appeal to a classical theorem of Jarník.

We begin by defining

$S_{N}(x,t)=\sum_{h=1}^{N}\frac{1}{h}\big{|}\sum_{n=1}^{N}e^{2\pi
ihn/N}e^{2\pi i(xn+tn^{2})}\big{|}=\sum_{h=1}^{N}\frac{1}{h}|w_{N}(x+h/N,t)|.$

Then we have the following ‘completion’ lemma proved in [CS] and [CS3].

######  Lemma 4.1 ([CS3], Lemma 2.2).

For all $(x,t)\in{\mathbb{T}}^{2}$ and any $1\leq N\leq M$ we have

$|w_{N}(x,t)|\lesssim|S_{M}(x,t)|.$

This lemma makes it easier to pass to dyadic scales, which is the main
motivation for working with $S_{N}$.

###  4.1. The upper bound

Recall that a Borel measure $\mu$ on $[0,1]$ is said to be $\beta$-dimensional
with $0<\beta\leq 1$ if

$c_{\beta}(\mu):=\sup_{I\subset{\mathbb{T}}}\frac{\mu(I)}{|I|^{\beta}}<\infty.$

Here the supremum is taken over intervals in $[0,1]$.

The upper bound in Proposition 1.1 will follow after combining Theorem 1.1
with the following two lemmas. The proof will be similar to arguments used to
study divergence sets for the Euclidean Schrödinger operator in [BBCR], [DZ],
[LR]. We recall that $\mathcal{H}^{\gamma}$ denotes $\gamma$-dimensional
Hausdorff measure.

######  Lemma 4.2 (Frostman’s Lemma).

Suppose $\mathcal{S}\subset[0,1]$ is a Borel set and
$\mathcal{H}^{\gamma}(S)>0$ with $\gamma>0$. Then there exists a nonzero
$\gamma$-dimensional Borel measure $\mu_{\gamma}$ supported on $\mathcal{S}$.

######  Lemma 4.3 ([EL]).

Fix $\epsilon>0$ and suppose $f\in H^{1/4+\epsilon}({\mathbb{T}})$ is a
function such that

$\big{\|}\sup_{0<t<1}|e^{it\Delta_{{\mathbb{T}}}}f|\|_{L^{4}({\mathbb{T}})}\lesssim_{\epsilon}\|f\|_{H^{1/4+\epsilon}}.$

Then for any $\gamma$-dimensional measure $\mu_{\gamma}$ on ${\mathbb{T}}$ we
also have

$\big{\|}\sup_{0<t<1}|e^{it\Delta_{{\mathbb{T}}}}f|\|_{L^{4}({\mathbb{T}},d\mu_{\gamma})}\lesssim
c_{\gamma}(\mu_{\gamma})^{1/4}\|f\|_{H^{s}},\ \ \ \ \
s>\frac{1}{2}-\frac{\gamma}{4}$

For a proof of Lemma 4.2 see [F] or [Ma]. Lemma 4.3 is a special case of the
proof of Proposition 5.2 in [EL].

We now fix $3/4\leq\alpha\leq 1$ and suppose that
$\mathcal{H}^{\gamma}({O}_{\alpha})>0$ for $\gamma>0$. Then by Lemma 4.2 there
exists a nonzero $\gamma$-dimensional Borel measure $\mu_{\gamma}$ supported
on $\mathcal{O}_{\alpha}$. Note that if we define

$\mathcal{O}_{\alpha}(N)=\\{x\in{\mathbb{T}}:\sup_{0<t<1}|w_{N}(x,t)|\geq
N^{\alpha}\\}$

then we have

(4.1) $\mathcal{O}_{\alpha}=\bigcap_{M\geq 1}\bigcup_{N\geq
M}\mathcal{O}_{\alpha}(N).$

We fix an integer $M>1$. For each $x\in\mathcal{O}_{\alpha}$ let

$N(x)=\min\\{N\geq M:\sup_{0<t<1}|w_{N}(x,t)|\geq N^{\alpha}\\}.$

Given $j\geq 1,$ define

$E_{j}=\\{x\in\mathcal{O}_{\alpha}:N(x)\in[2^{j-1}M,2^{j}M)\\}.$

The sets $E_{j}$ are pairwise disjoint by construction, and from (4.1) we have
$\mathcal{O}_{\alpha}\subset\bigcup_{j\geq 1}E_{j}$ for each $M\geq 1$. Note
that for each $x\in E_{j}$ we have

$1\leq N(x)^{-\alpha}\sup_{t}|w_{N(x)}(x,t)|\lesssim
2^{-j\alpha}M^{-\alpha}\sup_{t}S_{2^{j}M}(x,t),$

which follows from the definition of the $E_{j}$ and Lemma 4.1. Therefore

(4.2) $\mu_{\gamma}({\mathbb{T}})^{1/4}=\big{(}\int_{{\mathbb{T}}}1\
d\mu_{\gamma}(x)\big{)}^{1/4}\lesssim\big{(}\sum_{j=1}^{\infty}2^{-4\alpha
j}M^{-4\alpha}\int_{E_{j}}\sup_{0<t<1}|S_{2^{j}M}(x,t)|^{4}\
d\mu_{\gamma}(x)\big{)}^{1/4}.$

Recall that if $\mu$ is a $\gamma$-dimensional measure and $\mu^{t}$ is a
translate of $\mu$ then $\mu^{t}$ is also $\gamma$-dimensional, with
$c_{\gamma}(\mu^{t})=c_{\gamma}(\mu)$. We therefore conclude from Theorem 1.1
and Lemma 4.3 that for any $\epsilon>0$ we have

$\displaystyle\big{(}\int_{E_{j}}\sup_{0<t<1}|S_{2^{j}M}(x,t)|^{4}\
d\mu_{\gamma}(x)\big{)}^{1/4}$
$\displaystyle\leq\sum_{h=1}^{2^{j}M}\frac{1}{h}\big{(}\int_{E_{j}}\sup_{0<t<1}|w_{2^{j}M}(x+h/(2^{j}M),t)|^{4}\
d\mu_{\gamma}(x)\big{)}^{1/4}$ $\displaystyle\leq
C_{\epsilon}2^{j\epsilon}M^{\epsilon}2^{j(1-\gamma/4)}M^{1-\gamma/4}.$

Inserting this into (4.2) yields

(4.3)
$\mu_{\gamma}({\mathbb{T}})^{1/4}\lesssim_{\epsilon}M^{1-\alpha-\gamma/4+\epsilon}\big{(}\sum_{j=1}^{\infty}2^{-j(4\alpha-4+\gamma-\epsilon)}\big{)}^{1/4}.$

Let us now suppose that $\gamma>4(1-\alpha)$, say $\gamma=4(1-\alpha)+\delta$.
Then from (4.3) we get

(4.4)
$\mu_{\gamma}({\mathbb{T}})^{1/4}\lesssim_{\epsilon}M^{-\delta/4+\epsilon}\big{(}\sum_{j=1}^{\infty}2^{-j(\delta-\epsilon)})^{1/4}\lesssim_{\delta}M^{-\delta/8}$

if we choose $\epsilon=\delta/8$. But the implicit constant in (4.4) is
independent of $M$ and $M\geq 1$ was arbitrary, so we obtain a contradiction
by letting $M\rightarrow\infty$.

Therefore $\mathcal{H}^{\gamma}(\mathcal{O}_{\alpha})=0$ for
$\gamma>4(1-\alpha).$ It follows that the dimension of $\mathcal{O}_{\alpha}$
is less than or equal to $4(1-\alpha)$, completing the proof.

###  4.2. The lower bound

The argument in this case is similar to the approach used by Chen and
Shparlinski in [CS2], the main idea being that Weyl sums are known to be large
at points which are well-approximated by certain rational numbers. There are
also similar ideas used in [ChOs] to estimate the dimension of sets related to
the function $\sum_{n}\frac{1}{n}e^{2\pi i(xn+tn^{2})},$ as mentioned in
Section 1.2.

We will use the following theorem of Jarník that gives the dimension of sets
which are well-approximated by rationals.

######  Theorem 4.1 (Jarník).

Suppose $\beta>0$ and let

$\mathcal{J}_{\beta}=\\{x\in{\mathbb{T}}:|x-a/q|\leq q^{-(2+\beta)}\text{ for
infinitely many rationals }a/q\\}.$

Then the Hausdorff dimension of $\mathcal{J}_{\beta}$ is $\frac{2}{2+\beta}$.

See for example Theorem 10.3 in [F2] or [W] Section 9 for a proof. As observed
in [CS2], the argument in [F2] proves the the following stronger result.

######  Theorem 4.2.

Suppose $\beta>0$ and let

$\mathcal{J}^{\text{odd}}_{\beta}=\\{x\in{\mathbb{T}}:|x-a/q|\leq
q^{-(2+\beta)}\text{ for infinitely many rationals }a/q,\ q\text{ odd}\\}.$

Then the Hausdorff dimension of $\mathcal{J}^{\text{odd}}_{\beta}$ is
$\frac{2}{2+\beta}$.

In fact we can further restrict to primes $q$.

The following ‘major arc’ estimate makes clear the role of the Jarník theorem
in our analysis.

######  Proposition 4.4.

Fix $N>1$ and supose $1\leq q\leq N^{1/2}$ is an odd integer. Suppose
$a/q,b/q$ are rationals with $1\leq a,b<q$ and $(a,q)=1$. Suppose
$|x-b/q|\leq\frac{1}{100N}$ and $|t-a/q|\leq\frac{1}{100N^{2}}.$ Then

$\bigg{|}\sum_{n=1}^{N}e^{2\pi i(nx+n^{2}t)}\bigg{|}\geq c\frac{N}{q^{1/2}}.$

For a proof see [Oh], for example.

We fix a small parameter $\delta>0$. Let’s first suppose that $3/4<\alpha<1$.
Then if $q$ is an integer chosen large enough (depending only on $\alpha$) we
can find an integer $N_{q}$ such that

$100(N_{q}+1)>q^{\frac{1}{2(1-\alpha)}}>100N_{q}$

and

$q<N_{q}^{1/2}.$

Now suppose that $|x-a/q|\leq q^{-\frac{1}{2(1-\alpha)}}.$ Then we also have
$|x-a/q|<(100N_{q})^{-1}$, and so if $q$ is odd Proposition 4.4 implies that

$\sup_{0<t<1}|w_{N_{q}}(x,t)|\geq cq^{-1/2}N_{q}\geq
c^{\prime}N_{q}^{-1+\alpha}N_{q}\geq N_{q}^{\alpha-\delta}.$

We therefore conclude that

(4.5) $\mathcal{J}^{\text{odd}}_{\beta}\subset\mathcal{O}_{\alpha-\delta},\ \
\ \beta=\frac{4\alpha-3}{2(1-\alpha)}.$

Then from (4.5) and Theorem 4.2 it follows that

$\text{dim}(\mathcal{O}_{\alpha-\delta})\geq\text{dim}(\mathcal{J}^{\text{odd}}_{\beta})=\frac{2}{2+\beta},\
\ \ \ \beta=\frac{4\alpha-3}{2(1-\alpha)}.$

Now direct calculation shows that

$\frac{2}{2+\beta}=4(1-\alpha)$

and so we conclude that

$\text{dim}(\mathcal{O}_{\alpha-\delta})\geq 4(1-\alpha)$

for arbitrarily small $\delta>0$. It follows that if $3/4<\alpha<1$ and
$\delta>0$ is small enough we have

$\text{dim}(\mathcal{O}_{\alpha})\geq
4(1-(\alpha+\delta))=4(1-\alpha)-4\delta.$

The desired lower bound follows by letting $\delta\rightarrow 0$.

It remains to consider the endpoint case $\alpha=3/4$. But this follows from
the observation that $\mathcal{O}_{3/4}\supset\mathcal{O}_{3/4+\epsilon}$ for
any $\epsilon>0$, along with the fact that

$\dim(\mathcal{O}_{3/4+\epsilon})=1-4\epsilon$

as proven above.

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Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL
61801, USA

E-mail address<EMAIL_ADDRESS>