On Arithmetic Progressions in A plus B plus C

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.