Higher Distance Energies and Expanders with Structure

TL;DR

This paper introduces a novel application of higher moment energies from Additive Combinatorics to Discrete Geometry, resulting in improved bounds for distinct distances and expanding polynomial problems involving structured sets.

Contribution

It adapts higher moment energies to Discrete Geometry, providing new bounds and insights for problems with structured sets, and opens avenues for further research.

Findings

Improved bounds for distinct distances with local properties

Enhanced bounds for expanding polynomial problems with structured sets

Establishment of higher moment energies as a versatile tool in Discrete Geometry

Abstract

We adapt the idea of higher moment energies, originally used in Additive Combinatorics, so that it would apply to problems in Discrete Geometry. This new approach leads to a variety of new results, such as (i) Improved bounds for the problem of distinct distances with local properties. (ii) Improved bounds for problems involving expanding polynomials in ${\mathbb R}[x,y]$ (Elekes-Ronyai type bounds) when one or two of the sets have structure. Higher moment energies seem to be related to additional problems in Discrete Geometry, to lead to new elegant theory, and to raise new questions.