Levels of Distribution and the Affine Sieve

TL;DR

This paper provides an introductory overview of recent advances in sieve theory, focusing on the level of distribution and the affine sieve, highlighting key developments and their significance.

Contribution

It offers a clear exposition of the concepts of level of distribution and the affine sieve, connecting recent research to foundational ideas in sieve theory.

Findings

Summarizes Yitang Zhang's work on level of distribution.

Introduces the affine sieve method by Bourgain-Gamburd-Sarnak.

Highlights the importance of these topics in modern number theory.

Abstract

This article is an expanded version of the author's lecture in the Basic Notions Seminar at Harvard, September 2013. Our goal is a brief and introductory exposition of aspects of two topics in sieve theory which have received attention recently: (1) the spectacular work of Yitang Zhang, under the title "Level of Distribution," and (2) the so-called "Affine Sieve," introduced by Bourgain-Gamburd-Sarnak.