A lower bound for the size of a Minkowski sum of dilates

Abstract

Let A be a finite non-empty set of integers. An asymptotic estimate of the size of the sum of several dilates was obtained by Bukh. The unique known exact bound concerns the sum |A + k·A|, where k is a prime and |A| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author. Let k be an odd prime and assume that |A| > 8kk. A corollary to our main result states that |2·A + k·A|=(k+2)|A|-k2-k+2. Notice that |2·P+k·P|=(k+2)|P|-2k, if P is an arithmetic progression.