A Generalization of Roth's Theorem in Function Fields

Abstract

For n ∈ N = {1, 2, ...}, let D3([1, n]) denote the maximal cardinality of an integer subset of [1, n] containing no nontrivial 3-term arithmetic progression. In a fundamental paper [9], Roth proved that D3([1, n]) n/log log n. His result was later improved by Heath-Brown [4] and Szemerédi [11] to D3([1, n]) n/(log n)α for some small positive constant α > 0 (α = 1/20 in [11]). By introducing the notion of Bohr sets, Bourgain [2; 3] further improved this bound and showed that D3([1, n]) n(log log n)2 /(log n)2/3