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SUM-PRODUCT ESTIMATES FOR DIAGONAL MATRICES

Part of: Sequences and sets Polynomials and matrices

Published online by Cambridge University Press:  24 June 2020

AKSHAT MUDGAL Open the ORCID record for AKSHAT MUDGAL [Opens in a new window]
AKSHAT MUDGAL*
Affiliation:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK email am16393@bristol.ac.uk, amudgal@purdue.edu
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Abstract

Given $d\in \mathbb{N}$, we establish sum-product estimates for finite, nonempty 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, nonempty set of $d\times d$ diagonal matrices with real entries. Then, for all $\unicode[STIX]{x1D6FF}_{1}<1/3+5/5277$,

$$\begin{eqnarray}|A+A|+|A\cdot A|\gg _{d}|A|^{1+\unicode[STIX]{x1D6FF}_{1}/d},\end{eqnarray}$$
which strengthens a result of Chang [‘Additive and multiplicative structure in matrix spaces’, Combin. Probab. Comput.16(2) (2007), 219–238] in this setting.

Keywords

arithmetic combinatoricssum-product estimates

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 11C20: Matrices, determinants
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 28 - 37
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The author’s work was supported in part by a studentship sponsored by a European Research Council Advanced Grant under the European Union’s Horizon 2020 research and innovation programme via grant agreement no. 695223.

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