Tail Asymptotics for the Limiting Distribution of Theta Sums

dc.contributor.authorOsman, Tariqen
dc.contributor.departmentMathematics and Statisticsen
dc.contributor.supervisorCellarosi, Francesco
dc.date.accessioned2022-08-12T14:38:05Z
dc.date.available2022-08-12T14:38:05Z
dc.degree.grantorQueen's University at Kingstonen
dc.description.abstractWe 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.degreePhDen
dc.identifier.urihttp://hdl.handle.net/1974/30300
dc.language.isoengen
dc.relation.ispartofseriesCanadian thesesen
dc.rightsQueen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada*
dc.rightsProQuest PhD and Master's Theses International Dissemination Agreement*
dc.rightsIntellectual Property Guidelines at Queen's University*
dc.rightsCopying and Preserving Your Thesis*
dc.rightsThis 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.rightsAttribution 3.0 United States*
dc.rights.urihttp://creativecommons.org/licenses/by/3.0/us/*
dc.subjectProbability Theoryen
dc.subjectNumber Theoryen
dc.subjectDynamical Systemsen
dc.titleTail Asymptotics for the Limiting Distribution of Theta Sumsen
dc.typethesisen
Files
Original bundle
Now showing 1 - 1 of 1
Loading...
Thumbnail Image
Name:
Osman_Tariq_202208_PHD.pdf
Size:
2.78 MB
Format:
Adobe Portable Document Format
Description:
Thesis document
License bundle
Now showing 1 - 1 of 1
No Thumbnail Available
Name:
license.txt
Size:
1.67 KB
Format:
Item-specific license agreed upon to submission
Description: