Behrend's Theorem for Sequences Containing No k-Element Arithmetic Progression of a Certain Type

Let k and n be positive integers, and let d(n;k) be the maximum density in f0;1;2 : : : ;kn 1g of a set containing no arithmetic progression of k terms with first term a = ∑aik and common difference d = ∑εik, where 0 ai k 1, εi = 0 or 1, and εi = 1 ) ai = 0. Setting βk = limn!∞ d(n;k), we show that limk!∞ βk is either 0 or 1. Throughout, we shall use the notation [a;b) = fa;a+ 1;a+ 2; : : : ;b 1g, for nonnegative integers a < b. Also, if S is a set of nonnegative integers, then S(m) denotes jS\ [0;m)j. The upper asymptotic density of S will be denoted by d̄(S). Thus d̄(S) = limsup m!∞ m 1S(m): Similarly, the lower asymptotic density of S is d(S) = liminf m!∞ m 1S(m): Let rk(n) denote the largest cardinal of a subset A of [0;n) such that A contains no arithmetic progression of k terms, and let ρk = limn!∞ n rk(n). (This idea was introduced by Erdős, Turán, and Szekeres in [3], and then convergence of n rk(n) is shown in [2].) K. F. Roth [6] proved ρ3 = 0 in 1953 and E. Szemerédi [8] has shown that ρk = 0 for all k. Previous to these results, Felix Behrend [2] proved in 1937 that limk!∞ ρk equals either 0 or 1. In this paper we prove the analogous result where ρk is replaced by βk, the definition of βk being similar to that of ρk except that only arithmetic progressions of a certain type are considered. (At the time of this writing, the only known values for βk are β1 = β2 = 0.) The main idea for the proof is taken diresctly from Behrend’s paper. Definition. For each positive integer k, a k-diagonal is an arithmetic progression on k terms with first term a = ∑aik and common difference d = ∑εik, where for each i, 0 ai k 1, εi = 0 or 1, and εi = 1) ai = 0.