Related papers: Is an arbitrary diffused Borel probability measure…

In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's M\"{o}bius disjointness conjecture holds for any system for which every invariant Borel probability measure has…

Let $G$ be a noncompact connected Lie group, denote with $\rho$ a right Haar measure and choose a family of linearly independent left-invariant vector fields $\mathbf{X}$ on $G$ satisfying H\"ormander's condition. Let $\chi$ be a positive…

We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $\mu_G$, and a compact $A\subseteq G$ of positive measure with $\mu_G(A^2)\leq K\mu_G(A)$. Let $H$ be a closed normal subgroup of G and $\pi: G…

Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are…

We study the Minkowski problem corresponding to the p-harmonic measures and obtain results previously known for harmonic measures due to Jerison. We show that a class of Borel measures on spheres can be prescribed by p-harmonic measures on…

We study groups that can be defined as Polish, pro-countable groups, as non-archimedean groups with an invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product…

We investigate fixed point properties for isometric actions of topological groups on a wide class of metric spaces, with a particular emphasis on Hilbert spaces. Instead of requiring the action to be continuous, we assume that it is…

We show that the space of all bounded derivations from the disc algebra into its dual can be identified with the Hardy space $H^1$; using this, we infer that all such derivations are compact. Also, given a fixed derivation $D$, we construct…

For a homeomorphism $T$ on a compact metric space $X$, a $T$-invariant Borel probability measure $\mu$ on $X$ and a measure-theoretic quasifactor $\widetilde{\mu}$ of $\mu$, we study the relationship between the local entropy of the system…

Let $A$ be a compact set in ${\mathbb R}^p$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^s$ is the unique Borel probability measure with support in $A$ that minimizes $$…

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is…

We apply the theory of large-scale geometry of Polish groups to groups of absolutely continuous homeomorphisms. Let $M$ be either the compact interval or circle. We prove that the Polish group $\operatorname{AC}_+(M)$ of…

Let $\mu$ be a positive finite Borel measure on the unit circle. The associated Dirichlet space $\mathcal{D}(\mu)$ consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson…

We show a stability result for the recently established Brunn--Minkowski inequality in compact simple Lie groups. Namely, we prove that if two compact subsets $A, B$ of a compact simple Lie group $G$ satisfy $$ \mu(AB)^{1/d'} \leq (1 +…

All spaces are assumed to be separable and metrizable. Consider the following properties of a space $X$. (1) $X$ is Polish. (2) For every countable crowded $Q\subseteq X$ there exists a crowded $Q'\subseteq Q$ with compact closure. (3)…

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which…

We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In…

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1 \le p \neq n$. We prove that, for every pair of probability measures $\mu,\nu$ satisfying a…

Let $G$ be a compact Lie group of Lie type $B_{n},$ such as $SO(2n+1)$. We characterize the tuples\ $(x_{1},...,x_{L})$ of the elements $x_{j}\in G$ which have the property that the product of their conjugacy classes has non-empty interior.…

We begin the systematic study of decision problems for finitely generated groups given by a solution to their word problem. We relate this to the study of computable analysis on the space of marked groups. We point out that several distinct…