Sets of integers that do not contain long arithmetic progressions

Abstract

In 1946, Behrend gave a construction of dense finite sets of integers that do not contain 3-term arithmetic progressions. In 1961, Rankin generalized Behrend's construction to sets avoiding k-term arithmetic progressions, and in 2008 Elkin refined Behrend's 3-term construction. In this work, we combine Elkin's refinement and Rankin's generalization. Arithmetic progressions are handled as a special case of polynomial progressions. In 1946, Behrend gave a construction of dense finite sets of integers that do not contain a 3-term arithmetic progression (AP). In 1961, Rankin generalized Behrend's construction to sets avoiding k-term APs. In 2008, Elkin refined Behrend's 3-term construction, and later in 2008, Green & Wolf found a distinct approach (albeit morally similar) that is technically more straightforward. This work combines Elkin's refinement and Rankin's generalization in the Green & Wolf framework. A curious aspect of the construction is that we induct through sets that do not contain a long polynomial progression in order to construct a set without a long AP. The bounds for r_k(N), the largest size of a subset of {1,2,...,N} that does not contain a k element AP, are (where \log=\log_2, for sufficiently large N, with n=\ceiling{\log k}): r_3(N) > N (\sqrt{360}/(e π^{3/2})-ε) \sqrt[4]{2\log N} * 4^{-\sqrt{2 \log N}}, r_k(N) > CN 2^{-n 2^{(n-1)/2} \sqrt[n]{\log N}+\frac{1}{2n}\log\log N}. The improvement over earlier work is in the simplification of the construction, the explicitness of the bound for r_3, and in the \log\log term for general k.