Some applications of representation theory to the sum-product phenomenon

TL;DR

This paper introduces a novel approach using representation theory to estimate incidences, leading to new results in sum-product problems and applications to number theory conjectures.

Contribution

It presents a new method leveraging representation theory to improve incidence estimates and applies it to sums with multiplicative characters and Zaremba's conjecture.

Findings

New incidence estimation technique via representation theory

Applications to sums with multiplicative characters

Results related to Zaremba's conjecture

Abstract

In our paper, we introduce a new method for estimating incidences via representation theory. We obtain several applications to various sums with multiplicative characters and to Zaremba's conjecture from number theory.