相关论文: On the structure of distance sets over prime field…

We prove that if $E \subset {\mathbb F}_q^2$, $q \equiv 3 \mod 4$, has size greater than $Cq^{7/4}$, then $E$ determines a positive proportion of all congruence classes of triangles in ${\mathbb F}_q^2$. The approach in this paper is based…

We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean…

Let $H$ be an atomic monoid. The set of distances $\Delta (H)$ of $H$ is the set of all $d \in \mathbb{N}$ with the following property: there are irreducible elements $u\_1, \ldots, u\_k, v\_1 \ldots, v\_{k+d}$ such that $u\_1 \cdot \ldots…

We consider point sets in the $m$-dimensional affine space $\mathbb{F}_q^m$ where each squared Euclidean distance of two points is a square in $\mathbb{F}_q$. It turns out that the situation in $\mathbb{F}_q^m$ is rather similar to the one…

Let $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices…

For a set $E\subset \mathbb F_q^d$, we define the $k$-resultant magnitude set as $ \Delta_k(E) =\{\|\textbf{x}_1 + \dots + \textbf{x}_k\|\in \mathbb F_q: \textbf{x}_1, \dots, \textbf{x}_k \in E\},$ where $\|\textbf{v}\|=v_1^2+\cdots+ v_d^2$…

Every set of points $\mathcal{P}$ determines $\Omega(|\mathcal{P}| / \log |\mathcal{P}|)$ distances. A close version of this was initially conjectured by Erd\H{o}s in 1946 and rather recently proved by Guth and Katz. We show that when near…

We show that if $\mathcal{E}$ is a subset of the $d$-dimensional vector space over a finite field $\mathbbm{F}_q$ ($d \geq 3$) of cardinality $|\mathcal{E}| \geq (d-1)q^{d - 1}$, then the set of volumes of $d$-dimensional parallelepipeds…

Let $\mathcal{O}_K$ be the ring of integers of an algebraic number field $K$ embedded into $\mathbb{C}$. Let $X$ be a subset of the Euclidean space $\mathbb{R}^d$, and $D(X)$ be the set of the squared distances of two distinct points in…

Guth and Katz proved that any point set $\mathcal P$ in the plane determines $\Omega(|\mathcal P|/\log|\mathcal P|)$ distinct distances. We show that when near to this lower bound, a point set $\mathcal P$ of the form $A\times A$ must…

We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. A similar result takes place in the prime field F_p…

We obtain bounds on the number of triples that determine a given pair of dot products arising in a vector space over a finite field or a module over the set of integers modulo a power of a prime. More precisely, given $E\subset \mathbb…

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove…

In this short note, we study the distribution of spreads in a point set $\mathcal{P} \subseteq \mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $\varepsilon > 0$, if $|\mathcal{P}|…

Given a set of points $P \subset \mathbb F_q^2$ such that $|P|\geq q^{3/2}$ it is established that $|P|$ determines $\Omega(q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P| \geq q^{4/3}$, then for a positive…

We consider the problem of determining the number of distinct distances between two point sets in $\mathbb{R}^2$ where one point set $\mathcal{P}_1$ of size $m$ lies on a real algebraic curve of fixed degree $r$, and the other point set…

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, $k$-point configuration…

A Euclidean distance matrix $D(\alpha)$ is defined by $D_{ij}=(\alpha_i-\alpha_j)^2$, where $\alpha=(\alpha_1,\ldots,\alpha_n)$ is a real vector. We prove that $D(\alpha)$ cannot be written as a sum of $\left[2\sqrt{n}-2\right]$ nonnegative…

Let $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ be linearly disjoint number fields and let $\mathbb{Q}(\theta)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[\theta]$ may almost always be constructed in…

In this short note, we prove that for $\mathcal{E} \subset \mathbb{F}_q^d$ with $|\mathcal{E}| \geq q^{d-1} + O(q^2)$ then the set of determinants generated by $\mathcal{E}$ is $\mathbb{F}_q.$ This result is nearly optimal and generalizes…