Arithmetic progressions in sumsets of sparse sets

A set of positive integers A ⊂ Z_>0 is log-sparse if there is an absolute constant C so that for any positive integer x the sequence contains at most C elements in the interval [x, 2x). In this note we study arithmetic progressions in sums of log-sparse subsets of Z_>0. We prove that for any log-sparse subsets S₁, …, S_n of Z_>0, the sumset S = S₁ + · · · + S_n cannot contain an arithmetic progression of size greater than n^(1+o(1))n. We also show that this is nearly tight by proving that there exist log-sparse sets S₁, …, S_n such that S₁ +· · · +S_n contains an arithmetic progression of size n^(1−o(1))n.