A Note on Elkin’s Improvement of Behrend’s Construction

Abstract

Abstract. We provide a short proof of a recent result of Elkin in which large subsets of {1, . . . , N } free of 3-term progressions are constructed.

An Improved Construction of Progression-Free Sets

“…Since then several authors continued our line of research. Specifically, Green and Wolf [25] found a simpler proof of our result. They point out, however, that "the only advantage of our approach is brevity: it is based on ideas morally close to those of Elkin, and moreover, his argument is more constructive than ours."…”

“…They point out, however, that "the only advantage of our approach is brevity: it is based on ideas morally close to those of Elkin, and moreover, his argument is more constructive than ours." In an even more recent development O'Bryant [31] has combined our techniques with those of Rankin [33] and Green and Wolf [25], and improved Rankin's lower bound by a factor log ǫ n, for some small positive ǫ = ǫ(k). (In the preliminary version of our paper [19] we anticipated that our techniques could be useful to improve Rankin's bound.)…”

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

“…More precisely, the Erdős-Turán conjecture states that if δ and k are given, then there is a number N = N(k, δ) such that any set A ⊆ [1, N] with |A| ≥ δN contains a non-trivial k-AP. Roth [264] employed methods from Fourier analysis (or more specifically, the Hardy-Littlewood circle method) to prove the k = 3 case of the Erdős-Turán conjecture (see also [32,35,43,98,110,165,213,234,242,272,274,287,296]). Szemerédi [303] verified the Erdős-Turán conjecture for arithmetic progressions of length four.…”

“…See also [165] for a short proof of Elkin's result, and [242] for constructive lower bounds for r k (N). Schoen and Shkredov [277] using ideas from the paper of Sanders [273] and also the new probabilistic technique established by Croot and Sisask [98], obtained Behrend-type bounds for linear equations involving 6 or more variables.…”

Roth’s Theorem for Four Variables and Additive Structures in Sums of Sparse Sets

“…This theorem has been central to additive combinatorics, and improving the above bound has been the object of much research and has led to a wealth of interesting techniques being developed; see for example [4-6, 18, 23, 24, 27], to which we also refer for more history on the problem. However, it is not yet known whether r 3 (N ) C N /log N for some constant C; the current best upper bounds, due to Sanders [23] and Bloom [4], are of the form , as proved by Behrend [2] (but see also [12,17]). Now, most proofs of Roth's theorem easily extend to provide similar upper bounds for any translation invariant equation showed that much stronger bounds than those given above for r 3 (N ) hold for equations in six or more variables.…”