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

Authors:
Y. o. Hamidoune;J. RuÉ
Affiliations:
Upmc, université paris 06, 4 place jussieu, 75005 paris, france (e-mail: hamidoune@math.jussieu.fr);Lix, École polytechnique, 91128 palaiseau-cedex, france (e-mail: rue1982@lix.polytechnique.fr)
Venue:
Combinatorics, Probability and Computing
Year:
2011

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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.