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On Sums of Dilates

Published online by Cambridge University Press:  09 July 2009

JAVIER CILLERUELO ,
YAHYA O. HAMIDOUNE  and
ORIOL SERRA
JAVIER CILLERUELO
Affiliation:
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)
YAHYA O. HAMIDOUNE
Affiliation:
UPMC Université Paris 06, E. Combinatoire, Case 189, 4 Place Jussieu, 75005 Paris, France (e-mail: hamidoune@math.jussieu.fr)
ORIOL SERRA
Affiliation:
Universitat Politécnica de Catalunya, Jordi Girona, 1, E-08034 Barcelona, Spain (e-mail: oserra@ma4.upc.edu)
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Abstract

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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| − ⌈k(k + 2)/4⌉ where k · A = {ka: a ∈ A}. We also describe the sets for which equality holds.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 6 , November 2009 , pp. 871 - 880
Copyright
Copyright © Cambridge University Press 2009

References

[1]Bukh, B. (2008) Sums of dilates. Combin. Probab. Comput. 17 627–639.Google Scholar
[2]Cilleruelo, J., Silva, M. and Vinuesa, C. A sumset problem. Preprint.Google Scholar
[3]Nathanson, M. B. Inverse problems for linear forms over finite sets of integers. Available from: arXiv:0708.2304.Google Scholar