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

We consider quadratic Weyl sums $S_N(x;\alpha ,\beta )={\sum }_{n=1}^N\exp \text{\hspace{0.17em}⁣}\left[2\pi i((\frac{1}{2}{n}^2+\beta n)\text{\hspace{0.17em}⁣}x+\alpha n)\right]$ 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 $N\to \mathrm{\infty }$ 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 $S_N^f(x;\alpha ,\beta )={\sum }_{n\in \mathbb{Z}}f\left(\frac{n}{N}\right)\exp \text{\hspace{0.17em}⁣}\left[2\pi i((\frac{1}{2}{n}^2+\beta n)\text{\hspace{0.17em}⁣}x+\alpha n)\right]$ where $f$ is regular. The precise tails of the limiting distribution of $\frac{1}{N}{S}_N^{f_1}\overline{S_N^{f_2}}(x;\alpha ,\beta )$ as $N\to \mathrm{\infty }$ 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 $(\alpha ,\beta )\mathrm{\notin }{\mathbb{Q}}^2$ and $\alpha =\beta =0$ are considered.