Liu, Yu-RuZhao, Xiaomei2023-10-032023-10-032012-11https://doi.org/10.1307/mmj/1353098515http://hdl.handle.net/10012/19987Copyright © 2012 The University of MichiganFor 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)2/3enArticle