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Sumsets of semiconvex sets

Part of: Sequences and sets Designs and configurations

Published online by Cambridge University Press:  26 February 2021

Imre Ruzsa  and
Jozsef Solymosi
Imre Ruzsa*
Affiliation:
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Jozsef Solymosi
Affiliation:
University of British Columbia, Department of Mathematics, Vancouver, Canada e-mail: solymosi@math.ubc.ca
*
e-mail: ruzsa@renyi.hu
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Abstract

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the constant multiplier. We give a new proof to this result using bounds on crossing numbers of geometric graphs. We construct examples showing the limits of possible improvements. In particular, we show that there are arbitrarily large sets with different consecutive differences and sub-quadratic sumset size.

Keywords

sumset inequalitiesconvex sets of numbersadditive combinatorics

MSC classification

Primary: 05B10: Difference sets (number-theoretic, group-theoretic, etc.)
Secondary: 11B13: Additive bases, including sumsets 11B05: Density, gaps, topology
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 84 - 94
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Copyright
© Canadian Mathematical Society 2021

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