Sum-product estimates for diagonal matrices

TL;DR

This paper proves new sum-product estimates for finite sets of diagonal matrices in real space, showing that either the sum set or the product set must be significantly larger than the original set, strengthening previous results.

Contribution

The paper establishes quantitative sum-product bounds for sets of diagonal matrices, extending and strengthening prior results by Chang in this area.

Findings

Sum-product estimate for diagonal matrices with explicit bounds

Improved lower bounds on sum and product set sizes

Extension of Chang's result with quantitative strengthening

Abstract

Given $d \in \mathbb{N}$, we establish sum-product estimates for finite, non-empty subsets of $\mathbb{R}^d$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, non-empty set of $d \times d$ diagonal matrices with real entries. Then for all $\delta_1 < 1/3 + 5/5277$, we have \[ |A+A| + |A\cdot A| \gg_{d} |A|^{1 + \delta_{1}/d}. \] In this setting, the above estimate quantitatively strengthens a result of Chang.