Sumsets of Semiconvex sets

TL;DR

This paper studies the additive properties of semiconvex sets, providing a new proof for known bounds on sumsets, constructing examples that demonstrate the limits of these bounds, and exploring the structure of such sets.

Contribution

It introduces a novel proof technique using crossing number bounds and constructs examples showing the tightness and limitations of sumset size bounds for semiconvex sets.

Findings

New proof of sumset lower bounds using geometric graph crossing numbers

Construction of large semiconvex sets with sub-quadratic sumset sizes

Demonstration of the tightness of existing bounds and their limitations

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 sizes.