Erd\H{o}s and Arithmetic Progressions

TL;DR

This survey discusses Szemerédi's theorem, its implications, and the Erdős discrepancy problem, highlighting unresolved questions and speculative aspects in these areas of additive number theory.

Contribution

It provides a comprehensive overview of key results and open problems related to Szemerédi's theorem and the Erdős discrepancy problem as of 2013.

Findings

Overview of Szemerédi's theorem and its ramifications

Discussion of the Erdős discrepancy problem and related conjectures

Identification of open problems and areas for future research

Abstract

This is a short survey article written for the Erd\H{o}s centennial conference in Budapest in 2013. The main two topics covered are Szemer\'edi's theorem and its ramifications, and the Erd\H{o}s discrepancy problem. There is an emphasis on what we do not yet know, so much of the article is somewhat speculative.