Near arithmetic progressions in sparse sets

Abstract

Szemeredi’s theorem says that relatively dense sets contain arithmetic progressions. The purpose of this chapter is to present a result of Leth from Leth (Proc Am Math Soc 134:1579–1589, 2006) which shows that certain sparse sets contain “near” arithmetic progressions. We then detail the connection between the aforementioned theorem of Leth and the Erdős-Turán conjecture.