Related papers: Menger's covering property and groupwise density
Length density is a recently introduced factorization invariant, assigned to each element $n$ of a cancellative commutative atomic semigroup $S$, that measures how far the set of factorization lengths of $n$ is from being a full interval.…
We extend the notion of mex, which is central in combinatorial number theory, to an arbitrary combinatorial structure, and we prove a general theorem to determine the generating function of the objects having fixed mex. We then study this…
The cosmological dynamics of gravitational clustering satisfies an approximate invariance with respect to the cosmological parameters that is often used to simplify analytical computations. We describe how this approximate symmetry gives…
Abstract upper densities are monotone and subadditive functions from the power set of positive integers into the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper…
We prove the density hypothesis for wide families of arithmetic orbifolds arising from all division quaternion algebras over all number fields of bounded degree. Our power-saving bounds on the multiplicities of non-tempered representations…
In our paper we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq \mathbb{Z}/q\mathbb{Z}$ in the case of composite $q$. We obtain a quantitative version of a result of A. Fish about the structure of the…
The superconvergence phenomenon is shown for products of free, identically distributed random variables. We also show that a certain Holder regularity, first demonstrated by Biane for the density of a free additive convolution with a…
The article continues the study of the 'regular' arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification…
In this short review we introduce group field theory, a particular class of random tensor models, which represents nowadays one of the candidates for a fundamental theory of quantum gravity. We insist on the combinatorial richness of…
The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in "Moments of the Gaussian $\beta$ Ensembles and the large $N$ expansion of the densities" with the use of the topological recursion…
We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.
Regular variation of a multivariate measure with a Lebesgue density implies the regular variation of its density provided the density satisfies some regularity conditions. Unlike the univariate case, the converse also requires regularity…
It is shown that the probability density satisfies a hyperbolic equation of motion with the unique characteristic that in its many-particle form it contains derivatives acting at spatially remote regions. Based on this feature we explore…
We present a new lower bound on the Bowen-Radin maximal density of radius-R ball packings in the m-dimensional hyperbolic space, improving on the basic covering bound by factor \Omega(m(R+\ln m)) as m tends to infinity. This is done by…
We improve a recent result by giving the optimal conclusion possible both to the frequent universality criterion and the frequent hypercyclicity criterion using the notion of A-densities, where A refers to some weighted densities sharper…
We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of…
This short communication advances the hypothesis that the observed fractal structure of large-scale distribution of galaxies is due to a geometrical effect, which arises when observational quantities relevant for the characterization of a…
We explore recurrence properties arising from dynamical approach to combinatorial problems like the van der Waerden Theorem. We describe relations between these properties, study their consequences for dynamics, and demonstrate connections…
Without assuming the field structure on the additive group of real numbers $\mathbb{R}$ with the usual order $<,$ we explore the fact that every proper subgroup of $\mathbb{R}$ is either closed or dense. This property of subgroups of the…
We show that a convex body admits a translative dense packing in $\mathbb{R}^d$ if and only if it admits a translative economical covering.