Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters

TL;DR

This paper investigates the distributional behavior of quadratic Weyl sums with rational parameters, revealing conditions under which their scaled sums exhibit heavy tails or compact support, with explicit tail asymptotics and connections to theta functions.

Contribution

It characterizes the limiting distribution of quadratic Weyl sums with rational parameters, including explicit tail asymptotics and geometric interpretations, extending prior work to rational parameter cases.

Findings

Limiting distribution is heavy tailed or compactly supported depending on parameters.

Tail probability asymptotic to a constant times R^{-4} in the heavy tailed case.

Explicit description of tail behavior via measures related to theta functions.

Abstract

We consider quadratic Weyl sums $S_N(x;\alpha,\beta)=\sum_{n=1}^N \exp\!\left[2\pi i\left( \left(\tfrac{1}{2}n^2+\beta n\right)\!x+\alpha n\right)\right]$ for $(\alpha,\beta)\in\mathbb{Q}^2$, where $x\in\mathbb{R}$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. We prove that the limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;\alpha,\beta)$ as $N\to\infty$ is either heavy tailed or compactly supported, depending solely on $\alpha,\beta$. In the heavy tailed case, the probability (according to the limiting distribution) of landing outside a ball of radius $R$ is shown to be asymptotic to $\mathcal{T}(\alpha,\beta)R^{-4}$, where the constant $\mathcal{T}(\alpha,\beta)>0$ is explicit. The result follows from an analogous statement for products of generalized quadratic Weyl sums of the form $S_N^f(x;\alpha,\beta)=\sum_{n\in\mathbb{Z}} f\left(\frac{n}{N}\right)\exp\!\left[2\pi i\left( \left(\tfrac{1}{2}n^2+\beta n\right)\!x+\alpha n\right)\right]$ where $f$ is regular. The precise tails of the limiting distribution of $\frac{1}{N}S_N^{f_1}\bar{S_N^{f_2}}(x;\alpha,\beta)$ as $N\to\infty$ can be described in terms of a measure -- which depends on $(\alpha,\beta)$ -- of a super level set of a product of two Jacobi theta functions on a noncompact homogenous space. Such measures are obtained by means of an equidistribution theorem for rational horocycle lifts to a torus bundle over the unit tangent bundle to a cover of the classical modular surface. The cardinality and the geometry of orbits of rational points of the torus under the affine action of the theta group play a crucial role in the computation of $\mathcal{T}(\alpha,\beta)$. This paper complements and extends the works of Cellarosi and Marklof [6] and Marklof [32], where $(\alpha,\beta)\notin\mathbb{Q}^2$ and $\alpha=\beta=0$ are considered.