New Expander Bounds from Affine Group Energy

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 $A,B \subset \mathbb R$, $$|A+A| \ll K|A| \Rightarrow |AB| \gg_K |A||B|^{1/2+c}.$$