A lower bound for the size of a Minkowski sum of dilates
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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.
CitationHamidoune, Y.O., Rue, J. A lower bound for the size of a Minkowski sum of dilates. "Combinatorics probability and computing", 1 Març 2011, vol. 20, núm. 2, p. 249-256.