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Register a new userOn the number of dot product chains in finite fields and rings
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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 large finite set of points $E$, and a sequence of elements of the base field (or ring) $(alpha_1,ldots,alpha_k)$, we give conditions guaranteeing the expected number of $(k+1)$-tuples of distinct points $(x_1,dots, x_{k+1})in E^{k+1}$ satisfying $x_j cdot x_{j+1}=alpha_j$ for every $1leq j leq k$.
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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.
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.
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.
The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in the cyclic group case any pure Schur ring over R is the tensor product of a pure cyclotomic ring and Schur rings of rank 2 over non-fields. Moreover, it is shown that in contrast to the cyclic group case there are non-pure Schur rings over R that are not generalized wreath products.
Let $mathbb{F}_q$ be a finite field of order $q$, and $P$ be the paraboloid in $mathbb{F}_q^3$ defined by the equation $z=x^2+y^2$. A tuple $(a, b, c, d)in P^4$ is called a non-trivial energy tuple if $a+b=c+d$ and $a, b, c, d$ are distinct. For $Xsubset P$, let $mathcal{E}^+(X)$ be the number of non-trivial energy tuples in $X$. It was proved recently by Lewko (2020) that $mathcal{E}^+(X)ll |X|^{frac{99}{41}}$ for $|X|ll q^{frac{26}{21}}$. The main purposes of this paper are to prove lower bounds of $mathcal{E}^+(X)$ and to study related questions by using combinatorial arguments and a weak hypergraph regularity lemma developed recently by Lyall and Magyar (2020).
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