Sets whose sumset avoids a thin sequence

Let {a1,a2,a3,…} be an unbounded sequence of positive integers with an+1/an approaching α as n→∞, and let β>max(α,2). We show that for all sufficiently large x⩾0, if A⊂[0,x] is a set of nonnegative integers containing 0 and satisfying|A|⩾(1−1β)x, then we can represent some element of the sequence {an} as a pairwise sum of elements of A. We also prove an analogous result which holds for all x⩾0.