Sumsets containing infinite arithmetic progressions

AbstractLet A be a set of nonnegative integers such that dL(A) = w > 0. Let k be the least integer satisfying k ≥ 1w. It is proved that there is an infinite arithmetic progression with difference at most k + 1 such that every term of the progression can be written as a sum of exactly k2 − k distinct terms of A, and there is an infinite arithmetic progression with difference at most k2 − k such that every term of the progression can be written as a sum of exactly k + 1 distinct terms of A. A solution is also obtained to the infinite analog of a problem of Erdös and Freud on powers of 2 and on square-free numbers that can be represented as bounded sums of distinct elements chosen from a set A with positive density