Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T11:56:34.827Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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