Sum-product estimates for diagonal matrices

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 \[ |A+A| + |A\cdot A| \gg_{d} |A|^{1 + δ_{1}/d}. \] In this setting, the above estimate quantitatively strengthens a result of Chang.

A revised version will appear in Bulletin of the Australian Mathematical Society. 9 pages