Tail Asymptotics for the Limiting Distribution of Theta Sums
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Date
Authors
Osman, Tariq
Keyword
Probability Theory , Number Theory , Dynamical Systems
Abstract
We define theta sums to be exponential sums of the form S_{N}(x; \alpha, \beta) := \sum_{n =1}^{N} e((\tfrac{1}{2} n^2 + \beta n)x + \alpha n), where e(z) = e^{2 \pi i z}. If \alpha and \beta are fixed rational numbers, and x is chosen randomly from the unit interval, we use homogeneous dynamics to show that \tfrac{1}{N}S_{\lfloor sN\rfloor}S_{\lfloor tN\rfloor}, possesses a limiting distribution as N goes to infinity, for any s,t \in \mathbb{R}, and that this limiting distribution depends on the initial choice of \alpha and \beta. We then prove optimal tail asymptotics for the limiting distribution. More specifically, we prove that, according to the limiting distribution, the probability of landing outside a ball of sufficiently large radius $R$ for an explicit set of rational pairs (\alpha, \beta) is 0. For all other rational pairs (\alpha,\beta) we show that this probability is asymptotic to \tfrac{C_{\alpha,\beta}D_{s,t}}{\pi^2 R^4}(1 + O_{\varepsilon}(R^{-2 + \varepsilon})) for any \varepsilon > 0, where C_{\alpha,\beta} and D_{s,t} are explicit, positive constants. These results, in particular, imply that the limiting distribution of \tfrac{1}{N}S_{\lfloor sN\rfloor}S_{\lfloor tN\rfloor} when (\alpha,\beta) are rational, cannot be a Gaussian. This complements existing work of F. Cellarosi and J. Marklof when (\alpha,\beta) \in \mathbb{R}^2\setminus \mathbb{Q}^2, and completes the classification of the limiting tail behaviour of theta sums.
For the rational parameters that lead to compact support, we are able to prove a uniform bound for generalised theta sums S^f_N (x; \alpha,\beta) := \sum_{n\in \Z} f(\tfrac{n}{N}) e((\tfrac{1}{2} n^2 + \beta n)x + \alpha n), provided the weight function f is sufficiently regular.