Download this article
 Download this article For screen
For printing
Recent Issues
Volume 3, Issue 1
Volume 2, Issue 1
Volume 1, Issue 1
The Journal
About the Journal
Editorial Board
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN (electronic): 2834-4634
ISSN (print): 2834-4626
 
Other MSP Journals
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
Abstract

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}.

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