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

作者：Hamidoune Y O^*; Rue J

来源：Combinatorics Probability & Computing, 2011, 20(2): 249-256.

DOI：10.1017/S0963548310000520

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 vertical bar A + k . A vertical bar, where k is a prime and vertical bar A vertical bar 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 vertical bar A vertical bar > 8k(k). A corollary to our main result states that vertical bar 2 . A + k . A vertical bar >= (k + 2)vertical bar A vertical bar - k(2) - k + 2. Notice that vertical bar 2 . P + k . P vertical bar = (k + 2)vertical bar P vertical bar - 2k, if P is an arithmetic progression.

• 出版日期2011-3