On arithmetic progressions in A + B + C

TL;DR

This paper proves that the sumset of three subsets of integers with certain densities contains long arithmetic progressions, improving previous density bounds and utilizing advanced additive combinatorics techniques.

Contribution

It introduces new density thresholds for the existence of long arithmetic progressions in sumsets and applies recent methods from additive combinatorics and prime number theory.

Findings

Sumset A + B + C contains long arithmetic progressions under new density conditions.

Improves previous density bounds from ( ext{log} N)^{-1+o(1)} to ( ext{log} N)^{-2+ ext{epsilon}}.

Extends results to the setting of primes with new estimates.

Abstract

Our main result states that when A, B, C are subsets of Z/NZ of respective densities \alpha,\beta,\gamma, the sumset A + B + C contains an arithmetic progression of length at least e^{c(\log N)^c} for densities \alpha > (\log N)^{-2 + \epsilon} and \beta,\gamma > e^{-c(\log N)^c}, where c depends on \epsilon. Previous results of this type required one set to have density at least (\log N)^{-1 + o(1)}. Our argument relies on the method of Croot, Laba and Sisask to establish a similar estimate for the sumset A + B and on the recent advances on Roth's theorem by Sanders. We also obtain new estimates for the analogous problem in the primes studied by Cui, Li and Xue.