Related papers: Topological diagonalizations and Hausdorff dimensi…

A first-order expansion of the $\mathbb{R}$-vector space structure on $\mathbb{R}$ does not define every compact subset of every $\mathbb{R}^n$ if and only if topological and Hausdorff dimension coincide on all closed definable sets.…

It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs.…

We study the infimal value of the Hausdorff dimension of spaces that are H\"older equivalent to a given metric space; we call this bi-H\"older-invariant "H\"older dimension". This definition and some of our methods are analogous to those…

We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such…

We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical…

Let $B$ be a $d$-dimensional Gaussian process on $\mathbb{R}$, where the component are independents copies of a scalar Gaussian process $B_0$ on $\mathbb{R}_+$ with a given general variance function…

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can…

A `symbolic dynamical system' is a continuous transformation F:X-->X of a closed perfect subset X of A^V, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The…

For planar self-affine sets satisfying the strong separation condition, recent work of B\'ar\'any, Hochman, and Rapaport gives mild assumptions under which the Hausdorff dimension equals the affinity dimension. In this paper, we study…

For each prime p and a monic polynomial f, invertible over p, we define a group G_{p,f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group…

Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension dim(A) in [0,1] and a strong dimension Dim(A) in…

Hausdorff dimension of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections" "network" corresponding to a fractal set, $F$. This lead to the definition of the…

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable…

Let $g$ be a polynomial automorphism of $\C^2$. We study the Hausdorff dimension and topological dimension of the Julia set of $g$. We show that when $g$ is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly…

We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, $X$, into $\mathbb{R}^n$. The key question is `what is the generic dimension of $f(X)$?' and we consider two different…

This paper investigates the Hausdorff dimension properties of chains and antichains in Turing degrees and hyperarithmetic degrees. Our main contributions are threefold: First, for antichains in hyperarithmetic degrees, we prove that every…

We show that for the base two expansion \[ x=\sum_{i=1}^{\infty}2^{-(d_{1}(x)+d_{2}(x)+\dots+d_{i}(x))}\] with $x\in(0,1]$ and $d_{i}(x)\in\mathbb{N}$ the set $A=\{x|\lim_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension zero, this is…

We show that if $Y$ is a dense subspace of a Tychonoff space $X$, then $w(X)\leq nw(Y)^{Nag(Y)}$, where $Nag(Y)$ is the Nagami number of $Y$. In particular, if $Y$ is a Lindel\"of $\Sigma$-space, then $w(X)\leq nw(Y)^\omega\leq…

Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. Let $X$ be a smooth completion of $Y$ such that the boundary $X-Y$ is the support of an effective…