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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.
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|ge
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 Erd
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in the subobjec
We explore variants of ErdH os unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $mathbb F_q^d$ or $mathbb Z_q^d,$ where $q$ is a power of an odd prime. Specifically, given a
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.