On arithmetic progressions in A + B + C

2012/11/21 by Henriot, Kevin
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1211.4917

Abstract

Our main result states that when A, B, C are subsets of Z/NZ of respective densities α,β,γ, the sumset A + B + C contains an arithmetic progression of length at least e^{c(log N)c} for densities α> (log N)^{-2 + ε} and β,γ> e^{-c(log N)c}, where c depends on ε. 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.

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