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

Published online by Cambridge University Press:  06 December 2010

Y. O. HAMIDOUNE  and
J. RUÉ
Y. O. HAMIDOUNE
Affiliation:
UPMC, Université Paris 06, 4 Place Jussieu, 75005 Paris, France (e-mail: hamidoune@math.jussieu.fr)
J. RUÉ
Affiliation:
LIX, École Polytechnique, 91128 Palaiseau-CEDEX, France (e-mail: rue1982@lix.polytechnique.fr)
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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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 2 , March 2011 , pp. 249 - 256
Copyright
Copyright © Cambridge University Press 2010

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References

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