Related papers: Multi-dimensional Kronecker Sequences with a Small…
In this paper we prove in detail a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension $4$. We also present a new proof of the criterion for an algebraic continued fraction to have a proper…
In this paper, we classify Jordan superalgebras of dimension up to three over an algebraically closed field of characteristic different of two. Our main motivation to obtain such classification comes out from the intention to give an answer…
We estimate the number of incidences in a configuration of $m$ lines and $n$ points in dimension 3. The main term is $mn^{1/3}$ if we work over the real or complex numbers but $mn^{2/5}$ over finite fields. Both of these are optimal, aside…
In this paper we construct constant dimension space codes with prescribed minimum distance. There is an increased interest in space codes since a paper by Koetter and Kschischang were they gave an application in network coding. There is…
Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given $k$ there is a minimal diameter of subsets at which there exists a covering with…
The distances between flats of a Poisson $k$-flat process in the $d$-dimensional Euclidean space with $k<d/2$ are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a…
We prove supercongruences modulo $p^2$ for values of truncated hypergeometric series at some special points. The parameters of the hypergeometric series are $d$ copies of $1/2$ and $d$ copies of $1$ for any integer $d\ge2$.
We show that every multi-correlation sequence is the sum of a generalized nilsequence and a null-sequence. This proves a conjecture of N. Frantzikinakis. A key ingredient is the reduction of ergodic multidimensional inverse theorems to…
The most detailed constructions of microstate geometries, and particularly of superstrata, are done using $\mathcal{N} = (1,0)$ supergravity coupled to two anti-self-dual tensor multiplets in six dimensions. We show that an important…
Most well-known multidimensional continued fractions, including the M\"{o}nkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by…
It is proved that the Continuum Hypothesis implies that any sequence of rapid P-points of length $<{\mathfrak c}^{+}$ which is increasing with respect to the Rudin-Keisler ordering is bounded above by a rapid P-point. This is an improvement…
A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies…
We improve the status of the problem of determining minimum-sized percolating sets in $a \times b \times c$ grids under the $3$-neighbour process. Using several new constructions, we show that optimal percolating sets exist whenever…
The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to $p$-adic numbers where it presents many differences with…
We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of…
Consider two simple graphs, G1 and G2, with their respective vertex sets V(G1) and V(G2). The Kronecker product forms a new graph with a vertex set V(G1) X V(G2). In this new graph, two vertices, (x, y) and (u, v), are adjacent if and only…
We establish the upper bound in the multiplicity conjecture of Herzog, Huneke and Srinivasan for the codimension three almost complete intersections. We also give some partial results in the case where I is the aci linked to a complete…
We consider an infinite graph with the vertex set $\mathbb{Z}^2$ and edges connecting the vertices iff the Euclidean distance between the respective points is an integer, and the points do not lie on the same horizontal or vertical.…
We introduce the abstract notion of a chain, which is a sequence of $n$ points in the plane, ordered by $x$-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general…
In this paper we found an upper bound for the minimal quantifier depth of the first part of a monadic second-order sentence without asymptotic probability described by Jerzy Tyszkiewicz, which express the extension grid axiom in the…