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

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.