Correlations of sequences modulo one and statistics of geometrical objects associated to visible points

This thesis is divided into two major topics. In the first, we study the topic of distribution of sequences modulo one. In particular, we look at the spacing distributions between members of rational valued sequences modulo one. We come up with examples of many such sequences which behave as randomly chosen numbers from the unit interval. These include examples from the class of exponentially as well as sub-exponentially growing sequences. In the second part, we examine distribution questions for certain geometrical objects, for example, Farey-Ford and generalized Farey-Ford polygons and Farey-Ford parabolas associated to visible lattice points. As the names suggest, these objects are constructed based on the relation between visible points/Farey fractions and their geometrical interpretation in the form of Ford circles. We study the distribution of moments of various geometrical parameters associated to these objects by giving asymptotic formulas employing tools from analytic number theory.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Chaubey, Sneha

Contributors dc:contributor

• Zaharescu, Alexandru
• Boca, Florin
• Hildebrand, A. J.
• Robles, Nicolas

• Pair correlation
• Riemann zeta
• Visible lattice points

Statement dc:rights

• Copyright 2017 Sneha Chaubey

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/98203

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/98203