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A Lower Bound for the Size of a Minkowski Sum of Dilates

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.