On arithmetic progressions in A + B + C

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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