Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-11T20:29:42.400Z Has data issue: false hasContentIssue false
  • English
  • Français

An energy decomposition theorem for matrices and related questions

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

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: [email protected]
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: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anh, D. N. V., Ham, L. Q., Koh, D., Pham, T., and Vinh, L. A., On a theorem of Hegyvári and Hennecart . Pacific J. Math. 305(2020), no. 2, 407–421.CrossRefGoogle Scholar
Balog, A. and Wooley, T. D., A low-energy decomposition theorem . Q. J. Math. 68(2017), 207–226.Google Scholar
Hegyvári, N. and Hennecart, F., Expansion for cubes in the Heisenberg group . Forum Math. 30(2018), 227–236.CrossRefGoogle Scholar
Karabulut, Y. D., Koh, D., Pham, T., Shen, C.-Y., and Vinh, L. A., Expanding phenomena over matrix rings . Forum Math. 31(2019), 951–970.CrossRefGoogle Scholar
Mohammadi, A. and Stevens, S., Low-energy decomposition results over finite fields. Preprint, 2021. arXiv:2102.01655 Google Scholar
Murphy, B. and Petridis, G., Products of differences over arbitrary finite fields. Discrete Anal. 18(2018), 1–42.Google Scholar
Roche-Newton, O., Rudnev, M., and Shkredov, I. D., New sum-product type estimates over finite fields . Adv. Math. 293(2016), 589–605.CrossRefGoogle Scholar
Roche-Newton, O., Shparlinski, I. E., and Winterhof, A., Analogues of the Balog–Wooley decomposition for subsets of finite fields and character sums with convolutions . Ann. Comb. 23(2019), 183–205.CrossRefGoogle Scholar
Rudnev, M., On the number of incidences between points and planes in three dimensions . Combinatorica 38(2018), 219–254.CrossRefGoogle Scholar
Rudnev, M., Shkredov, I. D., and Stevens, S., On the energy variant of the sum-product conjecture . Rev. Mat. Iberoam. 36(2020), no. 1, 207–232.CrossRefGoogle Scholar
Sárközy, A., On sums and products of residues modulo $p$ $.$ Acta Arith. 118(2005), 403–409.CrossRefGoogle Scholar
Shkredov, I. D., An application of the sum-product phenomenon to sets avoiding several linear equations. Sbornik: Mathematics, 209(4), 580.CrossRefGoogle Scholar
Shkredov, I. D., A short remark on the multiplicative energy of the spectrum . Math. Notes 105(2019), 449–457.CrossRefGoogle Scholar
Shkredov, I. D., A remark on sets with small Wiener norm . In: Raigorodskii, A. and Rassias, M. T. (eds.), Trigonometric sums and their applications, Springer, Cham, 2020, pp. 261–272.CrossRefGoogle Scholar
Swaenepoel, C. and Winterhof, A., Additive double character sums over some structured sets and applications . Acta Arith. 199(2021), 135–143.CrossRefGoogle Scholar
Vu, V., Sum-product estimates via directed expanders . Math. Res. Lett. 15(2008), 375–388.CrossRefGoogle Scholar