New Expander Bounds from Affine Group Energy

Abstract: The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of Rudnev and Shkredov. We bound the energy of several systems of lines, in some cases obtaining a better energy bound than the corresponding bounds obtained by Rudnev and Shkredov by exploiting a connection with collinear quadruples. Our motivation for seeking to generalise and improve the incidence bound obtained by Rudnev and Shkredov comes from possible applications to sum-product problems. For example, we prove that, for any finite $A \subset \mathbb R$ the following superquadratic bound holds: [ \left| \left { \frac{ab-cd}{a-c} : a,b,c,d \in A \right } \right| \gg |A|^{2+\frac{1}{14}}. ] This improves the previously known bound with exponent $2$. We also give a threshold-beating asymmetric sum-product estimate for sets with small sum set by proving that there exists a positive constant $c$ such that for all finite $A,B \subset \mathbb R$, [ |A+A| \ll K|A| \Rightarrow |AB| \gg_K |A||B|^{1/2+c}. ]