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Sums of Dilates in p

Published online by Cambridge University Press:  30 October 2012

GONZALO FIZ PONTIVEROS
GONZALO FIZ PONTIVEROS*
Affiliation:
IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, Brasil (e-mail: gf232@cam.ac.uk)
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Abstract

We consider the problem of sums of dilates in groups of prime order. It is well known that sets with small density and small sumset in p behave like integer sets. Thus, given A ⊂ p of sufficiently small density, it is straightforward to show that

\begin{linenomath} $$| \lambda_{1}A+\lambda_{2}A+\cdots+ \lambda_{k}A | \ge\biggl(\sum_{i}|\lambda_{i}|\biggr)|A|- o(|A|).$$ \end{linenomath}
On the other hand, the behaviour for sets of large density turns out to be rather surprising. Indeed, for any ε > 0, we construct subsets of density 1/2–ε such that |A + λ A| ≤ (1–δ)p, showing that there is a very different behaviour for subsets of large density.

Keywords

Primary 11B1305B10
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 282 - 293
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Bukh, B. (2008) Sums of dilates. Combin. Probab. Comput. 17 627–639.CrossRefGoogle Scholar
[2]Cauchy, A. (1813) Recherches sur les nombres. J. École Polytechnique 9 99–116.Google Scholar
[3]Cilleruelo, J., Hamidoune, Y. O. and Serra, O. (2009) On sums of dilates. Combin. Probab. Comput. 18 871–880.CrossRefGoogle Scholar
[4]Cilleruelo, J., Silva, M. and Vinuesa, C. (2010) A sumset problem. J. Combin. Number Theory 2.Google Scholar
[5]Davenport, H. (1935) On the addition of residue classes. J. London Math. Soc. 10 30–32.CrossRefGoogle Scholar
[6]Freiman, G. (1973) Foundations of a Structural Theory of Set Addition, Vol. 37 of Translations of Mathematical Monographs, AMS.Google Scholar
[7]Green, B. and Ruzsa, I. (2006) Sets with small sumset and rectification. Bull. London Math. Soc. 38 43–52.CrossRefGoogle Scholar
[8]Hamidoune, Y. O. and Plagne, A. (2002) A generalization of Freiman's 3k-3 theorem. Acta Arithmetica 103 147–156.CrossRefGoogle Scholar
[9]Hamidoune, Y. O. and Rué, J. (2011) A lower bound for the size of a Minkowski sum of dilates. Combin. Probab. Comput. 20 249–256.CrossRefGoogle Scholar
[10]Heinemann, S. M. and Schmitt, O. (2001) Rokhlin's lemma for non-invertible maps. Dyn. Syst. Appl. 10 201–213.Google Scholar
[11]Nathanson, M. (2008) Inverse problems for linear forms over finite sets of integers. J. Ramanujan Math. Soc. 23 151–165.Google Scholar
[12]Plagne, A. (2011) Sums of dilates in groups of prime order. Combin. Probab. Comput. 20 867–873.CrossRefGoogle Scholar
[13]Rokhlin, V. A. (1963) Generators in ergodic theory. Vestnik Leningrad. Univ. Math. 18 26–32.Google Scholar
[14]Ruzsa, I. (1989) An application of graph theory to additive number theory. Scientia Ser. A 3 97–109.Google Scholar
[15]Ruzsa, I. (1996) Sums of finite sets. In Number Theory: New York Seminar, Springer, pp. 281–293.Google Scholar
[16]Tao, T. and Vu, V. (2006) Additive Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[17]Vosper, A. G. (1956) The critical pairs of subsets of a group of prime order. J. London Math. Soc. (2) 31 200–205.CrossRefGoogle Scholar