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We consider quadratic Weyl sums $S_N(x;\alpha,\beta)=\sum_{n=1}^N\exp\!\left[2\pi i\left( \left(\tfrac{1}{2}n^2+\beta n\right)\!x+\alphan\right)\right]$ for $(\alpha,\beta)\in\mathbb{Q}^2$, where $x\in\mathbb{R}$ israndomly distributed according to a probability measure absolutely continuouswith respect to the Lebesgue measure. We prove that the limiting distributionin 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 thelimiting distribution) of landing outside a ball of radius $R$ is shown to beasymptotic to $\mathcal{T}(\alpha,\beta)R^{-4}$, where the constant$\mathcal{T}(\alpha,\beta)>0$ is explicit. The result follows from an analogousstatement for products of generalized quadratic Weyl sums of the form$S_N^f(x;\alpha,\beta)=\sum_{n\in\mathbb{Z}}f\left(\frac{n}{N}\right)\exp\!\left[2\pi i\left( \left(\tfrac{1}{2}n^2+\betan\right)\!x+\alpha n\right)\right]$ where $f$ is regular. The precise tails ofthe limiting distribution of$\frac{1}{N}S_N^{f_1}\bar{S_N^{f_2}}(x;\alpha,\beta)$ as $N\to\infty$ can bedescribed in terms of a measure -- which depends on $(\alpha,\beta)$ -- of asuper level set of a product of two Jacobi theta functions on a noncompacthomogenous space. Such measures are obtained by means of an equidistributiontheorem for rational horocycle lifts to a torus bundle over the unit tangentbundle to a cover of the classical modular surface. The cardinality and thegeometry of orbits of rational points of the torus under the affine action ofthe theta group play a crucial role in the computation of$\mathcal{T}(\alpha,\beta)$. This paper complements and extends the works ofCellarosi and Marklof [6] and Marklof [32], where$(\alpha,\beta)\notin\mathbb{Q}^2$ and $\alpha=\beta=0$ are considered.

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