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Sumsets of semiconvex sets

Published online by Cambridge University Press:  26 February 2021

Imre Ruzsa*
Affiliation:
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Jozsef Solymosi
Affiliation:
University of British Columbia, Department of Mathematics, Vancouver, Canada e-mail: [email protected]

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.

Type
Article
Copyright
© Canadian Mathematical Society 2021

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