Sums, products and dilates on sparse graphs

Abstract

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