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On Sumsets of Convex Sets

Published online by Cambridge University Press:  07 July 2011

TOMASZ SCHOEN  and
ILYA D. SHKREDOV
TOMASZ SCHOEN
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland (e-mail: [email protected])
ILYA D. SHKREDOV
Affiliation:
Division of Algebra and Number Theory, Steklov Mathematical Institute, ul. Gubkina, 8, Moscow, 119991Russia (e-mail: [email protected])
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Abstract

A set of reals A = {a1,. . .,an} is called convex if ai+1 − ai > ai − ai−1 for all i. We prove, among other results, that for some c > 0 every convex A satisfies |A−A| ≥ c|A|8/5log−2/5|A|.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 793 - 798
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Elekes, G., Nathanson, M. and Ruzsa, I. Z. (2000) Convexity and sumsets. J. Number Theory 83 194–201.Google Scholar
[2]Garaev, M. Z. (2000) On lower bounds for the L 1-norm of exponential sums. Math. Notes 68 713–720.Google Scholar
[3]Garaev, M. Z. (2007) On an additive representation associated with the L 1-norm of an exponential sum. Rocky Mountain J. Math. 56 1551–1556.Google Scholar
[4]Garaev, M. Z. (2007) On the number of solutions of a diophantine equation with symmetric entries. J. Number Theory 125 201–209.CrossRefGoogle Scholar
[5]Garaev, M. Z. and Kueh, K.-L. (2005) On cardinality of sumsets. J. Aust. Math. Soc. 78 221–224.Google Scholar
[6]Hegyvári, N. (1986) On consecutive sums in sequences. Acta Math. Acad. Sci. Hungar. 48 193–200.CrossRefGoogle Scholar
[7]Iosevich, A., Konyagin, V. S., Rudnev, M. and Ten, V. (2006) Combinatorial complexity of convex sequences. Discrete Comput. Geom. 35 143–158.CrossRefGoogle Scholar
[8]Katz, N. H. and Koester, P. (2010) On additive doubling and energy. SIAM J. Discrete Math. 24 1684–1693.Google Scholar
[9]Konyagin, V. S. (2000) An estimate of the L 1-norm of an exponential sum. In The Theory of Approximations of Functions and Operators: Abstracts of Papers of the International Conference Dedicated to Stechkin's 80th Anniversary (in Russian), pp. 88–89.Google Scholar
[10]Sanders, T. (2010) On a non-abelian Balog–Szemerédi-type lemma. J. Aust. Math. Soc. 89 127–132.CrossRefGoogle Scholar
[11]Sanders, T. On Roth's theorem on progressions. Ann. of Math., to appear.Google Scholar
[12]Schoen, T. (2011) Near optimal bounds in Freiman's theorem. Duke Math. J. 158 1–12.Google Scholar
[13]Schoen, T. and Shkredov, I. D. Additive properties of multiplicative subgroups of . Quart. J. Math., to appear.Google Scholar
[14]Shkredov, I. D. and V'ugin, I. V. Additive shifts of multiplicative subgroups. Mat. Sbornik, to appear.Google Scholar
[15]Solymosi, J. (2009) Sumas contra productos. Gaceta de la Real Sociedad Matematica Española,12 (4).Google Scholar
[16]Szemerédi, E. and Trotter, W. T. (1983) Extremal problems in discrete geometry. Combinatorica 3 381–392.CrossRefGoogle Scholar