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.