Sumsets containing infinite arithmetic progressions (Q1095969)

The authors prove some quantitative results on infinite arithmetic progressions contained in sumsets of sets \(A\) (of nonnegative integers) of positive lower asymptotic density \(w\). If \(k\) is the smallest integer such that \(k\geq 1/w\), it is proved (i) that there is an infinite progression with difference at most \(k+1\) such that every term of the progression can be written as a sum of exactly \(k^2-k\) distinct terms of \(A\), (ii) there is an infinite arithmetic progression with difference at most \(k^2-k\) 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 shown to the infinite analog of two problems of Erdős and R. Freud on the representation of powers of 2 and square-free numbers as bounded sums of distinct elements chosen from a set with specified positive density.