Sums, products and dilates on sparse graphs

TL;DR

This paper establishes lower bounds on the sizes of sumsets and product sets constrained by sparse graphs, revealing new structural insights in additive combinatorics.

Contribution

It introduces novel bounds for sum and product sets on sparse graphs, extending classical results to graph-restricted sum and product operations.

Findings

Lower bounds on sumset and product set sizes in sparse graph settings

Demonstrates that at least one of the sets must be large relative to the graph size

Extends additive combinatorics results to graph-constrained operations

Abstract

Let $A \subset \mathbb R$ and $G \subset A \times A$. We prove that, for any $\lambda \in \mathbb R \setminus \{-1,0,1\}$, \[ \max \{|A+_G A|, |A+_G \lambda A|, |A\cdot_G A|\} \gg |G|^{6/11}. \]