A sum–product theorem in matrix rings over finite fields

Thang Pham

doi:10.1016/j.crma.2019.09.008

Number theory/Combinatorics

A sum–product theorem in matrix rings over finite fields
[Un théorème somme–produit dans les anneaux de matrices sur les corps finis]

Présenté par : the Editorial Board

Thang Pham ¹

¹ Department of Mathematics, University of Rochester, NY, USA

Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 766-770.

Résumé
Abstract

Dans cette Note, nous étudions le phénomène somme–produit dans les anneaux de matrices $M_{n} (F_{q})$ . Plus précisément, pour $A \subset M_{n} (F_{q})$ , nous montrons :

• si $| A \cap G L_{n} (F_{q}) | \leq | A | / 2$ , alors
$\max {| A + A |, | A A |} ≫ \min {| A | q, \frac{| A |^{3}}{q^{2 n^{2} - 2 n}}};$
• si $| A \cap G L_{n} (F_{q}) | \geq | A | / 2$ , alors
$\max {| A + A |, | A A |} ≫ \min {| A |^{\frac{2}{3}} q^{\frac{n^{2}}{3}}, \frac{| A |^{3 / 2}}{q^{\frac{n^{2}}{2} - \frac{1}{4}}}} .$

Nous donnons également une minoration de

| A + B |

pour

A \subset S L_{n} (F_{q})

B \subset M_{n} (F_{q})

In this note, we study a sum–product estimate over matrix rings $M_{n} (F_{q})$ . More precisely, for $A \subset M_{n} (F_{q})$ , we have

• if $| A \cap G L_{n} (F_{q}) | \leq | A | / 2$ , then
$\max {| A + A |, | A A |} ≫ \min {| A | q, \frac{| A |^{3}}{q^{2 n^{2} - 2 n}}};$
• if $| A \cap G L_{n} (F_{q}) | \geq | A | / 2$ , then
$\max {| A + A |, | A A |} ≫ \min {| A |^{\frac{2}{3}} q^{\frac{n^{2}}{3}}, \frac{| A |^{3 / 2}}{q^{\frac{n^{2}}{2} - \frac{1}{4}}}} .$

We also will provide a lower bound of

| A + B |

for

A \subset S L_{n} (F_{q})

and

B \subset M_{n} (F_{q})

Reçu le : 2018-12-22
Accepté le : 2019-09-12
Publié le : 2019-09-26

DOI : 10.1016/j.crma.2019.09.008

Affiliations des auteurs :

Thang Pham ¹

¹ Department of Mathematics, University of Rochester, NY, USA

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     author = {Thang Pham},
     title = {A sum{\textendash}product theorem in matrix rings over finite fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {766--770},
     publisher = {Elsevier},
     volume = {357},
     number = {10},
     year = {2019},
     doi = {10.1016/j.crma.2019.09.008},
     language = {en},
}

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Thang Pham. A sum–product theorem in matrix rings over finite fields. Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 766-770. doi : 10.1016/j.crma.2019.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.09.008/

Bibliographie
Cité par

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