Tail Asymptotics for the Limiting Distribution of Theta Sums

 dc.contributor.author Osman, Tariq en dc.contributor.department Mathematics and Statistics en dc.contributor.supervisor Cellarosi, Francesco dc.date.accessioned 2022-08-12T14:38:05Z dc.date.available 2022-08-12T14:38:05Z dc.degree.grantor Queen's University at Kingston en dc.description.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. en dc.description.degree PhD en dc.identifier.uri http://hdl.handle.net/1974/30300 dc.language.iso eng en dc.relation.ispartofseries Canadian theses en dc.rights Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada * dc.rights ProQuest PhD and Master's Theses International Dissemination Agreement * dc.rights Intellectual Property Guidelines at Queen's University * dc.rights Copying and Preserving Your Thesis * dc.rights 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. * dc.rights Attribution 3.0 United States * dc.rights.uri http://creativecommons.org/licenses/by/3.0/us/ * dc.subject Probability Theory en dc.subject Number Theory en dc.subject Dynamical Systems en dc.title Tail Asymptotics for the Limiting Distribution of Theta Sums en dc.type thesis en
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