3 results
An energy decomposition theorem for matrices and related questions
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- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 4 / December 2023
- Published online by Cambridge University Press:
- 15 May 2023, pp. 1280-1295
- Print publication:
- December 2023
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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
We also study some related questions on moderate expanders over matrix rings, namely, for
$$ \begin{align*} M(|A|) = \min\Bigg\{\,\frac{q^{4/3}}{|A|^{1/3}(\log|A|)^{2/3}},\, \frac{|A|^{4/5}}{q^{13/5}(\log|A|)^{27/10}}\,\Bigg\}. \end{align*} $$
$A, B, C\subseteq GL_2(\mathbb {F}_q)$, we have whenever
$$\begin{align*}|AB+C|, ~|(A+B)C|\gg q^4,\end{align*}$$
$|A||B||C|\gg q^{10 + 1/2}$. These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh ([2019], Expanding phenomena over matrix rings, 
$Forum Math.$, 31, 951–970).
A NOTE ON 2-LOCAL MAPS
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 49 / Issue 3 / October 2006
- Published online by Cambridge University Press:
- 25 January 2007, pp. 701-708
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On Traces of Separable Simple Sub Algebras in Matrix Rings
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- Journal:
- Canadian Mathematical Bulletin / Volume 38 / Issue 1 / 01 March 1995
- Published online by Cambridge University Press:
- 20 November 2018, pp. 104-111
- Print publication:
- 01 March 1995
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