# Questions tagged [hausdorff-dimension]

Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension.

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### Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube

This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...
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### Direct proof that the set of badly approximable numbers have full Hausdorff dimension without using Schmidt games

A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have $$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ .$$ The set of badly ...
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### On the set on which $|Df|$ is maximal for Lipschitz $f$

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with strict Lipschitz constant $L > 0$. That is, $|f(x) - f(y)| < L|x - y|$ for all $x \neq y$ in $\mathbb R^d$. Question: ...
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### Hausdorff measure

Let $(X,d)$ be a metric space and let $H^\alpha$ denote the $\alpha$-dimensional Hausdorff measure on $X$, where $\alpha$ is the Hausdorff dimension of $X$. Is there any simple condition on $X$ that ...
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### Hausdorff dimension of a compact Lie group [closed]

Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$. Now that $G$ is a metric space ...
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### Fraction dimensional "Euclidean" space

The “dimension” of Euclidean space $\mathbb{E}^n$ can be explained as an algebraic property, simply as a dimension of a vector space over the field $\mathbb{R}$. It also can be understood as a ...
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### Examples of probability measures with `fake' decay

To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ ...
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### A quantity that distinguishes finer than Hausdorff dimension

Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have ...
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### Hausdorff dimension of the graph of an increasing function

Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Let $\Gamma_f$ denote its graph. What can be said about the Hausdorff dimension of $\Gamma_f$? In ...
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### density of fractal measures

Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...
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Let $\Omega$ be the set of all infinite binary sequences $(x_i)_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$. Consider $0<\alpha<1$ and let $$K_\alpha=\{... 2 votes 0 answers 52 views ### Closed set containing infinite arithmetic progressions of ANY gap Let A\subseteq [0,\infty) be a set containing infinite arithmetic progressions of ANY gap, that is, for any d>0, there is t>0 such that t+kd\in A for all k\in \mathbb N. Molter and ... 1 vote 0 answers 204 views ### Compact sets of Hausdorff dimension zero I have a question about Hausdorff dimension. Suppose S is a compact subset of \mathbb{R}^n whose Hausdorff dimension is zero. Does it follow that S can be covered by a finite DISJOINT union of ... 0 votes 0 answers 116 views ### Hausdorff dimension of X\times X I am thinking of the following question: Let X\subseteq \mathbb R. Is it true that$$ \mathrm{dim_H}(X\times X)=2\mathrm{dim_H}(X)? $$My thoughts: We know that \mathrm{dim_H}(X)+\mathrm{dim_H}(... 9 votes 1 answer 411 views ### Is there a characterization of the Hausdorff measures? It is known that there is a unique measure on the Borel \sigma-algebra of \mathbb{R}^n such that the measure of the rectangle \prod_i [a_i,b_i[ is \prod_i (b_i-a_i). This is the Lebesgue ... 3 votes 1 answer 102 views ### volume entropy and Hausdoff dimension In relation to this question: Relation between volume entropy and Hausdorff dim of limit set? Given a metric space Z and a hyperbolic approximation X := hyp_{r_0}(Z) (as defined for example here).... 1 vote 1 answer 133 views ### Dimension of quotient of compact totally disconnected group action Assume that X is a compact metric space and G is compact totally disconnected group. And X has isometric free G-action i.e. gx=x\Rightarrow g=e. Then the following holds$${\rm dim}\ ...
Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0? I can think of something ad hoc like a Cantor middle $\frac13$ set where the ...
### Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...