On sets of integers not containing long arithmetic progressions

TL;DR

This paper constructs large subsets of the first N natural numbers that avoid long arithmetic progressions, extending Behrend's classical result for progressions of length 3 to longer lengths.

Contribution

It generalizes Behrend's construction to create large sets avoiding longer arithmetic progressions, with explicit bounds on their size.

Findings

Constructed subsets of size at least N exp(-C(log N)^{1/(k+1)})

Sets avoid arithmetic progressions of length 2^k+1

Extends classical results to longer progressions

Abstract

We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets which do not contain aritmetic progressions of length 3.