We consider quadratic Weyl sums \(S_N(x;\alpha ,\beta )=\sum _{n=1}^N \exp \!\left[ 2\pi i\left( \left( \tfrac{1}{2}n^2+\beta n\right) \!x\right. \right. \) \(\left. \left. +\alpha n\right) \right] \) 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 \(N\rightarrow \infty \) 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 \(\frac{1}{N}S_N^{f_1}\overline{S_N^{f_2}}(x;\alpha ,\beta )\) as \(N\rightarrow \infty \) 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 homogeneous 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 work complements and extends [6] and [33], in which the cases \((\alpha ,\beta )\notin \mathbb {Q}^2\) and \(\alpha =\beta =0\) are considered.
