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arXiv · 2005.13432

Sum-product estimates for diagonal matrices

Akshat Mudgal
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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 $δ_1 < 1/3 + 5/5277$, 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.

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Akshat Mudgal. 2020-05-27. Sum-product estimates for diagonal matrices. https://doi.org/10.1017/s000497272000060x

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