Improved Tail Estimates for the Distribution of Quadratic Weyl Sums

TL;DR

This paper improves the tail estimates for the distribution of quadratic Weyl sums, providing more precise asymptotic bounds for the probability of large deviations in both rational and irrational cases.

Contribution

It refines existing tail estimates for quadratic Weyl sums by developing improved techniques and making all constants explicit, applicable to both rational and irrational cases.

Findings

Enhanced tail bounds with explicit constants

Applicable to rational and irrational quadratic Weyl sums

Utilizes equidistribution of rational horocycle lifts

Abstract

We consider quadratic Weyl sums $S_N(x;c,\alpha)=\sum_{n=1}^N\exp\{2\pi i((\frac{1}{2}n^2+cn)x+\alpha n)\}$ for $c=\alpha=0$ (the rational case) or $(c,\alpha)\notin\mathbb{Q}^2$ (the irrational case), where $x$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. The limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;c,\alpha)$ as $N\to\infty$ was described by Marklof [13] (respectively Cellarosi and Marklof [5]) in the rational (resp. irrational) case. According to the limiting distribution, the probability of landing outside a ball of radius $R$ is known to be asymptotic to $\frac{4\log 2}{\pi^2}R^{-4}(1+o(1))$ in the rational case and to $\frac{6}{\pi^2}R^{-6}(1+O(R^{-12/31}))$ in the irrational case, as $R\to\infty$. In this work we refine the technique of Cellarosi and Marklof [5] to improve the known tail estimates to $\frac{4\log 2}{\pi^2}R^{-4}(1+O_\varepsilon(R^{-2+\varepsilon}))$ and $\frac{6}{\pi^2}R^{-6}(1+O_\varepsilon(R^{-2+\varepsilon}))$ for every $\varepsilon>0$. In the rational case, we rely on the equidistribution of a rational horocycle lift to a torus bundle over the unit tangent bundle to the classical modular surface. All the constants implied by the $O_\varepsilon$-notations are made explicit