Behrend's theorem for sequences containing no k-element arithmetic progression of a certain type

Abstract

Let k and n be positive integers, and let d(n, k) be the maximum density in {0, 1, 2,…, kⁿ − 1} of a set containing no arithmetic progression of k terms with first term a = Σ a_ikⁱ and common difference d = Σ ϵ_ikⁱ, where 0 ⩽ a_i ⩽ k − 1, ϵ_i = 0 or 1, and ϵ_i = 1 ⇒ a_i = 0. Setting β_k = lim_n→∞d(n, k), we show that lim_k→∞β_k is either 0 or 1.