Discretized sum-product estimates in matrix algebras

TL;DR

This paper extends Bourgain's discretized sum-product theorem from scalar fields to the setting of matrix algebras, broadening the scope of sum-product phenomena in algebraic structures.

Contribution

The work introduces a generalized discretized sum-product estimate applicable to matrix algebras, expanding previous scalar-based results to higher-dimensional algebraic systems.

Findings

Established a sum-product estimate in matrix algebras

Demonstrated the theorem's applicability to higher-dimensional structures

Extended Bourgain's theorem beyond scalar fields

Abstract

We generalize Bourgain's discretized sum-product theorem to matrix algebras.