Some sum-product estimates in matrix rings over finite fields

Abstract

We study some sum-product problems over matrix rings. Firstly, for A,B,C⊆Mn(Fq), we have|A+BC|≳qn2, whenever |A||B||C|≳q3n2−n+12. Secondly, if a set A in Mn(Fq) satisfies |A|≥C(n)qn2−1 for some sufficiently large C(n), then we havemax⁡{|A+A|,|AA|}≳min⁡{|A|2qn2−n+14,qn2/3|A|2/3}. These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.