Number theory/Combinatorics
A sum–product theorem in matrix rings over finite fields
Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 766-770.

In this note, we study a sum–product estimate over matrix rings $Mn(Fq)$. More precisely, for $A⊂Mn(Fq)$, we have

• • if $|A∩GLn(Fq)|≤|A|/2$, then $max⁡{|A+A|,|AA|}≫min⁡{|A|q,|A|3q2n2−2n};$
• • if $|A∩GLn(Fq)|≥|A|/2$, then $max⁡{|A+A|,|AA|}≫min⁡{|A|23qn23,|A|3/2qn22−14}.$
We also will provide a lower bound of $|A+B|$ for $A⊂SLn(Fq)$ and $B⊂Mn(Fq)$.

Dans cette Note, nous étudions le phénomène somme–produit dans les anneaux de matrices $Mn(Fq)$. Plus précisément, pour $A⊂Mn(Fq)$, nous montrons :

• • si $|A∩GLn(Fq)|≤|A|/2$, alors $max⁡{|A+A|,|AA|}≫min⁡{|A|q,|A|3q2n2−2n};$
• • si $|A∩GLn(Fq)|≥|A|/2$, alors $max⁡{|A+A|,|AA|}≫min⁡{|A|23qn23,|A|3/2qn22−14}.$
Nous donnons également une minoration de $|A+B|$ pour $A⊂SLn(Fq)$ et $B⊂Mn(Fq)$.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.09.008

Thang Pham 1

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

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