Sumsets of semiconvex sets

Ruzsa, Z. Imre and Solymosi, József (2022) Sumsets of semiconvex sets. CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 65 (1). pp. 84-94. ISSN 0008-4395

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Abstract

We investigate additive properties of sets <![CDATA[ $A,$ ]]> where <![CDATA[ $A=\\{a_1,a_2,\\ldots,a_k\\}$ ]]> is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that <![CDATA[ $|A+B|\\geq c|A||B|^{1/2}$ ]]> for any finite set of numbers <![CDATA[ $B.$ ]]> 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. © Canadian Mathematical Society 2021.

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