# Tagged Questions

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### Controlling the size of the balls in Hausdorff dimension/measure

Let $X$ be a compact metric space, and let $$\nu_s(X):=\sup\limits_{\varepsilon>0} \inf\limits_{\mathcal{E}} \sum\limits_{E \in \mathcal E} \mathrm{diam}(E)^s$$ be the $s$-dimensional Hausdorff ...
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### Hausdorff Dimension of Exceptional Set for Carleson's Theorem

In Mattila's book Fourier Analysis and Hausdorff Dimension, Mattila presents a result of Barcelo, Bennett, Carbery, and Rogers about convergence of solutions of the Schrodinger equation to the initial ...
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### Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?

I am interested in the Hausdorff dimension of the Apollonian circle packing. There seem to be two numerical calculations of the value: 1.305686729(10) from P.B ...
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### When is Hausdorff measure a Frostman measure?

Let $(X,d)$ be a metric space and let $\mathcal{H}^s$ be the $s$-dimensional Hausdorff measure on $X$. For a measure $\mu$ on $X$, we say that $\mu$ is a Frostman measure (sometimes referred as ...
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### Packing measure and Kleinian groups

There has been "some" debate on the notion of fractal (as an illustration, see for example the discussion in this link). One of the possible notions includes relating Hausdorff dimension and packing ...
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### Jarník-Besicovitch and outer measure

The set $A_\tau$ of irrational numbers $x$ which are $\tau$-approximable, i.e., that satisfy the estimate $$\left|x - \frac{p}{q}\right| \leq \frac{1}{q^\tau}$$ for infinitely many rationals $p/q$, ...
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### Hausdorff dimension of a Cantor-like set

Suppose $K$ is a subset of $[0,1]$ with the following property: for almost $x,y \in K$, we have $$\frac{x+y}{2} \not\in K.$$ (Here, "almost in $K$" means "in $K$ except for a countable subset"). ...
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### Best Hölder exponents of surjective maps from the unit square to the unit cube

The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...
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### How can dimension depend on the point?

Let $M$ be a metric space. For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension. For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
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### Geometric measures different from Hausdorff

$\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$ In geometric measure theory there are various notions of $m$-dimensional measure for sets $A\subset \RR^n$...
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### Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?

I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero. In the 1966 paper "Additive gruppen mit vorgegebener ...
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### Can Hausdorff dimension make sets into a Tropical Semiring?

If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : \...
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### Hausdorff measure of the zero set

Let $f : \mathbb R^n\to \mathbb R$ continuous, for which there exist $x,y\in\mathbb R^n$, such that $f(x)f(y)<0$. Is it true that the Hausdorff dimension of the zero set of $f$ is at least $n-1$?
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### Are there any good techniques for calculating Hausdorff measure?

I'm aware that many techniques have been developed for the purpose of calculating Hausdorff dimension (although I'm fairly unfamiliar with them), but my question is whether or not we have any good ...
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