Related papers: On the strength of Hausdorff's gap condition

For a stationary set S subseteq omega_1 and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S,…

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $\Gamma\backslash G$, where $G$ is any connected…

We investigate connections between resolvability and different forms of tightness. This study is adjacent to [1,2]. We construct a non-regular refinement $\tau^*$ of the natural topology of the real line $\mathbb{R}$ with properties such…

In this paper, we prove the identity $\dim_{\textrm H}(F)=d\cdot \dim_{\textrm H}(\alpha^{-1}(F))$, where $\dim_{\textrm H}$ denotes Hausdorff dimension, $F\subseteq \mathbb{R}^d$, and $\alpha:[0,1]\to [0,1]^d$ is a function whose…

We continue here [She88] but we do not rely on it. The motivation was a conjecture of Galvin stating that 2^{omega} >= omega_2 + omega_2-> [omega_1]^{n}_{h(n)} is consistent for a suitable h: omega-> omega. In section 5 we disprove this and…

The partition function of an N=2 gauge theory in the Omega-background satisfies, for generic value of the parameter beta=-eps_1/eps_2, the, in general extended, but otherwise beta-independent, holomorphic anomaly equation of special…

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…

A famous result of Hausdorff states that a sphere with countably many points removed can be partitioned into three pieces A,B,C such that A is congruent to B (i.e., there is an isometry of the sphere which sends A to B), B is congruent to…

Heisenberg antiferromagnets in a strong uniform magnetic field $H$ are expected to exhibit a gapless phase with a global O(2) symmetry. In many real magnets, a small energy gap is induced by additional interactions that can be viewed as a…

In this paper, we study the uniform random covering problem in general metric space $(X,d)$. Let $\omega=(\omega_n)_{n\in\mathbb N}$ be a sequence of independent identically distributed random variables on $(X,\mu)$, and…

Let $X = G/\Gamma$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\operatorname{Ad}$-diagonalizable over $\mathbb{C}$ and whose action on $(X,m_X)$ is mixing. In this…

We continue our study of outer elements of the noncommutative H^p spaces associated with Arveson's subdiagonal algebras. We extend our generalized inner-outer factorization theorem, and our characterization of outer elements, to include the…

We introduce Hausdorff (complexity) classes, which provide canonical characterizations of the intermediate levels of the iterated exponential hierarchies, including the Polynomial Hierarchy, the (Weak) Exponential Hierarchy, and…

Assuming Jensen's principle diamond, there is a compact Hausdorff space X which is hereditarily Lindelof, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing…

Let $f\colon X\rightarrow Y$ be a continuous surjection of compact Hausdorff spaces. By $$f_*\colon\mathfrak{M}(X)\rightarrow\mathfrak{M}(Y),\ \mu\mapsto \mu\circ f^{-1} \quad{\rm and}\quad 2^f\colon2^X\rightarrow2^Y,\ A\mapsto f[A]$$ we…

In this article, we show that in a $Q$-doubling space $(X,d,\mu),$ $Q>1,$ that supports a $Q$-Poincar\'e inequality and satisfies a chain condition, sets of $Q$-capacity zero have generalized Hausdorff $h$-measure zero for…

We establish that constructive continued fraction dimension originally defined using $s$-gales is robust, but surprisingly, that the effective continued fraction dimension and effective (base-$b$) Hausdorff dimension of the same real can be…

Let $A\sub \R^{n+r}$ be a set definable in an o-minimal expansion $\S$ of the real field, $A' \sub \R^r$ be its projection, and assume that the non-empty fibers $A_a \sub \R^n$ are compact for all $a \in A'$ and uniformly bounded, {\em…

We address facts and open questions concerning the degree of ill-posedness of the composite Hausdorff moment problem aimed at the recovery of a function $x \in L^2(0,1)$ from elements of the infinite dimensional sequence space $\ell^2$ that…

A topological space $X$ is $strongly$ $rigid$ if each non-constant continuous map $f:X\to X$ is the identity map of $X$. A Hausdorff topological space $X$ is called $Brown$ if for any nonempty open sets $U,V\subseteq X$ the intersection…