Abstract

Let r(k, N) be the maximal cardinality of a subset A of {1, 2,..., N} which does not contain an arithmetic progression of length k. That is, A does not contain a subset of the form {x + jy: 0 ≤ j < k}, where x, y are integers with y ̸ = 0. Erdös and Turan [3] initiated the study of these quantities in 1936. In particular they conjectured that r(k, N) = o(N) for all k, that is every set of integers of positive asymptotic density contains arbitrarily long arithmetic progressions. In 1953, Roth [8] showed that r(3, N) = o(N). The Erdös–Turan conjecture was verified by Szemerédi [11, 12], a result with a very broad influence. Subsequently, rather different proofs of Szemerédi’s theorem were given by Furstenberg [4] and Gowers [5, 6]. Gowers’s proof provides, for the first time, upper bounds on r(k, N) given by a bounded tower of exponentials. An intriguing question of Erdös asks if r(3, N) ≤ CN/(log N) 1+δ for some positive δ. Bourgain’s article [2] contains the best current upper bound of CN √ log log N on r(3, N). log N In this article we are interested in the converse question of finding large subsets of {1,..., N} which do not contain arithmetic progressions. Behrend, in 1946, [1] (building on earlier work of Salem and Spencer [10]) considered three term arithmetic progressions, and showed that r(3, N) ≥ N exp(−C √ log N). The purpose of this paper is to show that if one considers longer arithmetic progressions then Behrend’s estimate can be further improved as follows.

Citations