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Sums of Dilates in Groups of Prime Order

Published online by Cambridge University Press:  06 October 2011

ALAIN PLAGNE
ALAIN PLAGNE*
Affiliation:
Centre de Mathématiques Laurent Schwartz, École polytechnique, 91128 Palaiseau Cedex, France (e-mail: plagne@math.polytechnique.fr)
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Abstract

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or −1 and if ⊂ ℤ/pℤ is not too large (with respect to p), then | + t ⋅ | is significantly larger than 2|| (unless |t| = 3). In the important case |t| = 2, we obtain for instance | + t ⋅ | ≥ 2.08 ||−2.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 6 , November 2011 , pp. 867 - 873
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Bukh, B. (2008) Sums of dilates. Combin. Probab. Comput. 17 627–639.Google Scholar
[2]Cauchy, A.-L. (1813) Recherches sur les nombres. J. École Polytechnique 9 99–123.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]Du, S.-S., Cao, H.-Q. and Sun, Z.-W. On a sumset problem for the integers. arXiv:1011.5438Google Scholar
[6]Freiman, G. A. (1973) Foundations of a Structural Theory of Set Addition, Vol. 37 of Translations of Mathematical Monographs, AMS.Google Scholar
[7]Hamidoune, Y. O. and Plagne, A. (2002) A generalization of Freiman's 3k − 3 theorem. Acta Arith. 103 147–156.Google Scholar
[8]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.Google Scholar
[9]Lev, V. F. (2005) Distribution of points on arcs. Integers: Electron. J. Combin. Number Theory 5 #A11.Google Scholar
[10]Lev, V. F. (2007) More on points and arcs. Integers: Electron. J. Combin. Number Theory 7 #A24.Google Scholar
[11]Ljujic, Z. A lower bound for the size of a sum of dilates. arXiv.1101.5425Google Scholar
[12]Nathanson, M. (2008) Inverse problems for linear forms over finite sets of integers. J. Ramanujan Math. Soc. 23 151–165.Google Scholar