Tail Asymptotics for the Limiting Distribution of Theta Sums

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.

Department dc:contributor.department

Mathematics and Statistics

Author dc:creator

• Osman, Tariq

Advisor dc:contributor.supervisor

• Cellarosi, Francesco

• Probability Theory
• Number Theory
• Dynamical Systems

Statement dc:rights

• Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada
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• Intellectual Property Guidelines at Queen's University
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• This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.
• Attribution 3.0 United States

Licence dc:rights.uri

• http://creativecommons.org/licenses/by/3.0/us/

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1974/30300

OAI identifier oai:identifier

oai:queensu.scholaris.ca:1974/30300