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Some sum-product estimates in matrix rings over finite fields (2107.03788v1)
Published 8 Jul 2021 in math.CO and math.NT
Abstract: We study some sum-product problems over matrix rings. Firstly, for $A, B, C\subseteq M_n(\mathbb{F}_q)$, we have $$ |A+BC|\gtrsim q{n2}, $$ whenever $|A||B||C|\gtrsim q{3n2-\frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(\mathbb{F}_q)$ satisfies $|A|\geq C(n)q{n2-1}$ for some sufficiently large $C(n)$, then we have $$ \max{|A+A|, |AA|}\gtrsim \min\left{\frac{|A|2}{q{n2-\frac{n+1}{4}}}, q{n2/3}|A|{2/3}\right}. $$ 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.