An Improved Construction of Progression-Free Sets

TL;DR

This paper introduces a new, elementary construction that improves the lower bound for the size of dense, progression-free subsets of integers, surpassing Behrend's long-standing result from 1946.

Contribution

The authors present a novel, elementary construction that enhances Behrend's lower bound for progression-free sets by a factor of Theta(\sqrt{ ext{log n}}), demonstrating the previous bound was not optimal.

Findings

Improved lower bound for progression-free sets by a factor of Theta(\sqrt{ ext{log n}})

Construction is elementary and self-contained

Shows Behrend's construction is not optimal

Abstract

The problem of constructing dense subsets S of {1,2,..,n} that contain no arithmetic triple was introduced by Erdos and Turan in 1936. They have presented a construction with |S| = \Omega(n^{\log_3 2}) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is |S| = Omega({n \over {2^{2 \sqrt{2} \sqrt{\log_2 n}} \cdot \log^{1/4} n}}). Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend was reported since 1946. In this paper we present a construction that improves the result of Behrend by a factor of Theta(\sqrt{\log n}), and shows that |S| = Omega({n \over {2^{2 \sqrt{2} \sqrt{\log_2 n}}}} \cdot \log^{1/4} n). In particular, our result implies that the construction of Behrend is not optimal. Our construction and proof are elementary and self-contained.