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Four-term progression free sets with three-term progressions in all large subsets

Pohoata, Cosmin and Roche-Newton, Oliver (2019) Four-term progression free sets with three-term progressions in all large subsets. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20200110-152805209

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Abstract

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such progression. We prove that there exists a positive constant c and a set A ⊂ F^n_q which does not contain a four-term arithmetic progression, with the property that for every subset A′ ⊂ A with |A′| ≥ |A|^(1−c), A′ contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth-type theorem in random subsets of F^n_q, which improves a result of Kohayakawa-Luczak-Rödl/Tao-Vu. We also discuss a similar phenomenon over the integers, where we show that for all ϵ > 0, and all sufficiently large N ∈ N, there exists a four-term progression-free set A of size N with the property that for every subset A′ ⊂ A with |A′| ≫ 1/(logN)^(1−ϵ)⋅ N contains a nontrivial three term arithmetic progression. Finally, we include another application of our methods, showing that for sets in F^n_q or Z the property of "having nontrivial three-term progressions in all large subsets" is almost entirely uncorrelated with the property of "having large additive energy".


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1905.08457arXivDiscussion Paper
Record Number:CaltechAUTHORS:20200110-152805209
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200110-152805209
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100645
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Jan 2020 00:53
Last Modified:11 Jan 2020 00:53

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