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

Chaubey, Sneha

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Description

Title

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

Author(s)

Chaubey, Sneha

Issue Date

2017-07-10

Director of Research (if dissertation) or Advisor (if thesis)

Zaharescu, Alexandru

Doctoral Committee Chair(s)

Boca, Florin

Committee Member(s)

• Hildebrand, A. J.
• Robles, Nicolas

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Pair correlation
• Riemann zeta
• Visible lattice points

Abstract

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.

Graduation Semester

2017-08

Type of Resource

text

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Copyright and License Information

Copyright 2017 Sneha Chaubey

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