SUMSETS CONTAINING A TERM OF A SEQUENCE

Let S = {s(1), s(2),...} be an unbounded sequence of positive integers with s(n+1)/s(n) approaching alpha as n -> infinity and let ss > max(alpha, 2). We show that for all sufficiently large positive integers l, if A. [0, l] with l is an element of A, gcdA = 1 and |A| >= (2 - k/ lambda ss)l/(lambda + 1), where lambda = [k/ss], then kA boolean AND S not equal theta for 2 < ss <= 3 and k >= 2 ss/(ss - 2) or for ss > 3 and k >= 3.