Combinatorics, Probability and Computing

Paper

On Sums of Dilates

JAVIER CILLERUELOa1, YAHYA O. HAMIDOUNEa2 and ORIOL SERRAa3

a1 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Madrid, Spain (e-mail: franciscojavier.cilleruelo@uam.es)

a2 UPMC Université Paris 06, E. Combinatoire, Case 189, 4 Place Jussieu, 75005 Paris, France (e-mail: hamidoune@math.jussieu.fr)

a3 Universitat Politécnica de Catalunya, Jordi Girona, 1, E-08034 Barcelona, Spain (e-mail: oserra@ma4.upc.edu)

Abstract

For k prime and A a finite set of integers with |A| ≥ 3(k − 1)2(k − 1)! we prove that |A + k · A| ≥ (k + 1)|A| − xs2308k(k + 2)/4xs2309 where k · A = {ka: a xs2208 A}. We also describe the sets for which equality holds.

(Received February 27 2009)

(Revised May 26 2009)

(Online publication July 09 2009)

Footnotes

Supported by Project MTM2008-03880 from MYCIT (Spain) and the joint Madrid Region-UAM project TENU3 (CCG08-UAM/ESP-3906).