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

Let A be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size 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.