Tail Asymptotics for Generalised Theta Sums with Rational Parameters

Abstract: We define generalised theta sums as exponential sums of the form S^f_N(x; \alpha, \beta) := \sum_{n \in \mathbb Z} f(n/N) e((1/2 n^2 + \beta n)x + \alpha n), where e(z) = e^{2 \pi i z}. If \alpha and \beta are fixed real numbers, and x is chosen randomly from the unit interval, we may use homogeneous dynamics to show that N^{-1/2} S^f_N$ possesses a limiting distribution as N goes to infinity, provided f is sufficiently regular. In joint work with F. Cellarosi, we prove that for specific rational pairs (\alpha, \beta) this limiting distribution is compactly supported and that all other rational pairs lead to a limiting distribution with heavy tails. This complements the existing work of F. Cellarosi and J. Marklof where at least one of \alpha or \beta is irrational.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar