# New Expander Bounds from Affine Group Energy

## TL;DR

This paper develops new bounds on affine group energy using incidence geometry, leading to improved sum-product estimates and revealing deeper structural properties of finite sets.

## Contribution

It introduces novel energy bounds for systems of lines via affine group analysis, enhancing previous results and enabling stronger sum-product inequalities.

## Key findings

- Improved energy bounds for systems of lines using affine group techniques.
- A superquadratic lower bound for a specific ratio set derived from finite sets.
- A threshold-beating sum-product estimate for sets with small sum sets.

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