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

• Published in: arXiv.org 2023-01
• Main Authors: Cellarosi, Francesco, Osman, Tariq
• Format: Article
• Language: English
• Subjects: Constraining; Sums; Toruses
• ISSN: 2331-8422

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.