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The Kelley–Meka bounds for sets free of three-term arithmetic progressions

Thomas F. Bloom and Olof Sisask

Vol. 2 (2023), No. 1, 15–44

We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if A {1,,N} has no nontrivial three-term arithmetic progressions then |A| exp (c(log N)112)N, where c > 0 is a constant.

Although our proof is identical to that of Kelley and Meka in all of the main ideas, we also incorporate some minor simplifications relating to Bohr sets. This eases some of the technical difficulties tackled by Kelley and Meka and widens the scope of their method. As a consequence, we improve the lower bounds for the problem of finding long arithmetic progressions in A + A + A, where A {1,,N}.

additive combinatorics, arithmetic progressions
Mathematical Subject Classification
Primary: 11B25, 11B30
Received: 20 February 2023
Revised: 21 November 2023
Accepted: 26 November 2023
Published: 31 December 2023
Thomas F. Bloom
Mathematical Institute
University of Oxford
United Kingdom
Olof Sisask
Department of Mathematics
Stockholm University