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Register a new userSome sum-product estimates in matrix rings over finite fields
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We study some sum-product problems over matrix rings. Firstly, for $A, B, Csubseteq M_n(mathbb{F}_q)$, we have $$ |A+BC|gtrsim q^{n^2}, $$ whenever $|A||B||C|gtrsim q^{3n^2-frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(mathbb{F}_q)$ satisfies $|A|geq C(n)q^{n^2-1}$ for some sufficiently large $C(n)$, then we have $$ max{|A+A|, |AA|}gtrsim minleft{frac{|A|^2}{q^{n^2-frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}right}. $$ 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.
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In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $Asubset mathbb{F}_q$ we have [|(A-A)^2+(A-A)^2|gg |A|^{1+frac{1}{21}}.] This can be viewed as the ErdH{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that [max{|A+A|, |A^2+A^2|}gg |A|^{1+frac{1}{42}}, ~|A+A^2|gg |A|^{1+frac{1}{84}}.]
In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in cite{HI07} and cite{HIKR07}. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions. In particular, our machinery enables us to prove that if $E subset {Bbb F}_q^d$, $d ge 4$, the $d$-dimensional vector space over a finite field ${Bbb F}_q$, of size much greater than $q^{frac{d}{2}}$, and if $E$ is a product set, then the set of volumes of $d$-dimensional parallelepipeds determined by $E$ covers ${Bbb F}_q$. This result is sharp as can be seen by taking $E$ to equal to $A times A times ... times A$, where $A$ is a sub-field of ${Bbb F}_q$ of size $sqrt{q}$. In three dimensions we establish the same result if $|E| gtrsim q^{{15/8}}$. We prove in three dimensions that the set of volumes covers a positive proportion of ${Bbb F}_q$ if $|E| ge Cq^{{3/2}}$. Finally we show that in three dimensions the set of volumes covers a positive proportion of ${Bbb F}_q$ if $|E| ge Cq^2$, without any further assumptions on $E$, which is again sharp as taking $E$ to be a 2-plane through the origin shows.
In this paper, we study the expanding phenomena in the setting of higher dimensional matrix rings. More precisely, we obtain a sum-product estimate for large subsets and show that x+yz, x(y+z) are moderate expanders over the matrix ring, and xy + z + t is strong expander over the matrix rings. These results generalize recent results of Y.D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author.
In this paper, we study dot-product sets and $k$-simplices in vector spaces over finite rings. We show that if $E$ is sufficiently large then the dot-product set of $E$ covers the whole ring. In higher dimensional cases, if $E$ is sufficiently large then the set of simplices and the set of dot-product simplices determined by $E$, up to congurence, have positive densities.
Let $mathcal{S}$ be a finite cyclic semigroup written additively. An element $e$ of $mathcal{S}$ is said to be idempotent if $e+e=e$. A sequence $T$ over $mathcal{S}$ is called {sl idempotent-sum free} provided that no idempotent of $mathcal{S}$ can be represented as a sum of one or more terms from $T$. We prove that an idempotent-sum free sequence over $mathcal{S}$ of length over approximately a half of the size of $mathcal{S}$ is well-structured. This result generalizes the Savchev-Chen Structure Theorem for zero-sum free sequences over finite cyclic groups.
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