Description

Let x_n be a sequence of real numbers between 0 and 1. Such a sequence is called equidistributed if for every open interval (a,b)⊆ (0,1), the proportion of {x_n, 1≤ n ≤ N} lying in (a,b) tends to b-a as N tends to infinity. Problem of determining when an arbitrary sequence equidistributes is an interesting question in analytic number theory. In this project we will learn about various results of this genre. Results on exponential sums are vital tool in solving these problems. We will start by learning about the Weyl's criterion for equidistribution in one and higher dimensions, and its application to the equidistribution of {nα:α irrational}. We will then learn about Weyl's inequality and equidistribution of the sequence {n^kα} for an irrational α. We will also consider equidistribution modulo N and application to Roth's theorem.

Prerequisites

Resources

Davenport H. Analytic Methods for Diophantine Equations and Diophantine Inequalities, (Chapter 3), Cambridge university press, ISBN: 9780521605830.

Editors: Granville A., Rudnick Z. Equidistribution in Number Theory, An Introduction, (Pages 1-11), Springer, Nato science series, volume 237.

Walkden C. Lecture notes on Ergodic theory, Lecture VI, Manchester University Lecture notes