Related papers: On the volume set of point sets in vector spaces o…

We present here quantitative versions in 1 dimension of Faltings'theorem according to which the set of the K-rational points (where K is a given number field) of an abelian variety A definied over K, which are close (with respect to a…

Any compact body in ${\mathbb R}^N$ with smooth boundary defines a two-valued function on the space of affine hyperplanes: the volumes of two parts into which these hyperplanes cut the body. This function is never algebraic if $N$ is even…

Let $q$ be a prime power and $\mathbb{F}_q$ be a finite field with $q$ elements. Let $e$ and $d$ be positive integers. In this paper, for $d\geq2$ and $q\equiv1(\mathrm{mod}~ed(d-1))$, we calculate the number of points on an algebraic curve…

Motivated by the well-known Paley graphs over finite fields and their generalizations, in this paper we explore a natural multiplicative-additive analogue of such graphs arising from vector spaces over finite fields. Namely, if $n\ge 2$ and…

In this paper we introduce a graph structure, called subspace sum graph $\mathcal{G}(\mathbb{V})$ on a finite dimensional vector space $\mathbb{V}$ where the vertex set is the collection of non-trivial proper subspaces of a vector space and…

The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements--all in analogy with formulas for finite sets (which are the special case of q=1). A direct-sum decomposition of a…

Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well…

The classical Dvoretzky--Rogers lemma provides a deterministic algorithm by which, from any set of isotropic vectors in Euclidean $d$-space, one can select a subset of $d$ vectors whose determinant is not too small. Subsequently,…

We study the first mixed volume for nonconvex sets and apply the results to limits of discrete isoperimetric problems. Let $ M,N \subset \mathbb{R}^d$. Define $D_N (M)=\lim_{\epsilon \downarrow 0} \frac{|M+\epsilon N|-|M|}{\epsilon}$…

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to…

In this paper, we study the analogous Erd\H{o}s similarity conjecture in higher dimensions and generalize the Eigen-Falconer theorem. We show that if $A=\{\boldsymbol{x}_n\}_{n=1}^\infty \subseteq \mathbb{R}^d$ is a sequence of non-zero…

Federer's characterization states that a set $E\subset \mathbb{R}^n$ is of finite perimeter if and only if $\mathcal H^{n-1}(\partial^*E)<\infty$. Here the measure-theoretic boundary $\partial^*E$ consists of those points where both $E$ and…

We prove that in any finite set of $\mathbb Z^d$ with $d\ge 3$, there is a subset whose capacity and volume are both of the same order as the capacity of the initial set. As an application we obtain estimates on the probability of {\it…

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We…

We investigate the nodal volume of random hyperspherical harmonics $\lbrace T_{\ell;d}\rbrace_{\ell\in \mathbb N}$ on the $d$-dimensional unit sphere ($d\ge 2$). We exploit an orthogonal expansion in terms of Laguerre polynomials; this…

Let $\mathcal{X} \subset \mathbb{P}_k^d$ be Drinfeld's halfspace over a finite field $k$ and let $\mathcal{E}$ be a homogeneous vector bundle on $\mathbb{P}_k^d$. The paper deals with two different descriptions of the space of global…

Let $\mathcal{Q}_1$ and $\mathcal{Q}_2$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^n(\mathbb{F}_q)$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result…

Some geometric facts concerning sets with positive reach in the Euclidean $d$-dimensional space ${\bf E}^d$ are proved. For $x_1$ and $x_2$ in ${\bf E}^d$ and $R>0$ let us denote by ${\mathfrak H}(x_1,x_2,R)$ the intersection of all closed…

There are two positive, absolute constants $c_{1}$ and $c_{2}$ so that the volume of the difference set of the $d$-dimensional Euclidean ball and an inscribed polytope with n vertices is larger than $$ c_{2}\ d\…

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a…