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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 $1leq j leq k$.
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
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 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
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
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 $Xsu