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• W3021221530 abstract "Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every $epsilon > 0$ there is some $N_0(epsilon)$ such that for every $N ge N_0(epsilon)$ and $A subset [N]$ with $|A| = alpha N$, there is some nonzero $d$ such that $A$ contains at least $(alpha^3 - epsilon) N$ three-term arithmetic progressions with common difference $d$. We prove that the minimum $N_0(epsilon)$ in Green's theorem is an exponential tower of 2s of height on the order of $log(1/epsilon)$. Both the lower and upper bounds are new. It shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary." @default.
• W3021221530 created "2020-05-13" @default.
• W3021221530 creator A5006789469 @default.
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• W3021221530 date "2020-04-28" @default.
• W3021221530 modified "2023-09-23" @default.
• W3021221530 title "Tower-type bounds for Roth's theorem with popular differences" @default.
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• W3021221530 doi "https://doi.org/10.48550/arxiv.2004.13690" @default.
• W3021221530 hasPublicationYear "2020" @default.
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