Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-11T10:32:08.304Z Has data issue: false hasContentIssue false
  • English
  • Français

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: schoen@amu.edu.pl)
ILYA D. SHKREDOV
Affiliation:
Division of Algebra and Number Theory, Steklov Mathematical Institute, ul. Gubkina, 8, Moscow, 119991Russia (e-mail: ilya.shkredov@gmail.com)
Get access Rights & Permissions [Opens in a new window]

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

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]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