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An energy decomposition theorem for matrices and related questions

Part of: Finite fields and commutative rings (number-theoretic aspects) Special matrices

Published online by Cambridge University Press:  15 May 2023

Ali Mohammadi ,
Thang Pham Open the ORCID record for Thang Pham [Opens in a new window]  and
Yiting Wang
Ali Mohammadi
Affiliation:
School of Mathematics and Statistics, University of Sydney, Camperdown, NSW 2006, Australia e-mail: ali.mohammadi.np@gmail.com
Thang Pham*
Affiliation:
University of Science, Vietnam National University, Hanoi 100000, Vietnam
Yiting Wang
Affiliation:
Institute of Science and Technology Austria, Klosterneuburg 3400, Austria e-mail: yiting.wang@ist.ac.at
*
e-mail: phamanhthang.vnu@gmail.com
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Abstract

Given $A\subseteq GL_2(\mathbb {F}_q)$, we prove that there exist disjoint subsets $B, C\subseteq A$ such that $A = B \sqcup C$ and their additive and multiplicative energies satisfying

$$\begin{align*}\max\{\,E_{+}(B),\, E_{\times}(C)\,\}\ll \frac{|A|^3}{M(|A|)}, \end{align*}$$

where

$$ \begin{align*} M(|A|) = \min\Bigg\{\,\frac{q^{4/3}}{|A|^{1/3}(\log|A|)^{2/3}},\, \frac{|A|^{4/5}}{q^{13/5}(\log|A|)^{27/10}}\,\Bigg\}. \end{align*} $$
We also study some related questions on moderate expanders over matrix rings, namely, for $A, B, C\subseteq GL_2(\mathbb {F}_q)$, we have
$$\begin{align*}|AB+C|, ~|(A+B)C|\gg q^4,\end{align*}$$
whenever $|A||B||C|\gg q^{10 + 1/2}$. These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh ([2019], Expanding phenomena over matrix rings, $Forum Math.$, 31, 951–970).

Keywords

Matrix ringsexpanderssum-product estimatesenergy estimatesfinite fields

MSC classification

Primary: 11T06: Polynomials
Secondary: 15B33: Matrices over special rings (quaternions, finite fields, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 4 , December 2023 , pp. 1280 - 1295
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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