# 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. To be used with [tag:fractals] or [tag:dimension-theory].

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### Can the fractal dimension of a coastline be less than 1?

I am currently investigating the fractal dimension of the coastline of the island the Palm Jumeirah, including the crescent. Using the Hausdorff method I have reached an answer of 0.879. This is less ...
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### Hausdorff dimension of sum of sets

Assume $0<r_0<n$. Are there sets $A,B\subseteq \mathbb{R}^n$, such that the Hasudorff dimension of $A,B$ are zero, But $\dim_H(A+B)=r_0$? When $r_0$ is integer, I have found(By attention to page ...
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### the intermediate property for topological Hausdorff dimension

According to page $889$, Theorem $3.6$ of A new fractal dimension..., the topological Hausdorff dimension of a subset $X$ of $\mathbb{R}^n$(or a subset of a separable metric space), can be defined as ...
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### Complement of strictly less than (n-1)-dimensional set is connected?

Let $n\in \mathbb{N}$ and $A\subseteq \mathbb{R}^n$ be a subset such that its Hausdorff dimension is strictly less than $n-1$. Is it then true that $\mathbb{R}^n\setminus A$ is (path-)connected? ...
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### Hausdorff dimension from above

Let $A_n$ be a set of Hausdorff dimension $1-\frac{1}{n}$ then, the set $$A=\cup_n A_n$$ Has Hausdorff dimension $1$ (nevertheless having $H_1(A)=0$). My question is: can we do the same thing from ...
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### Hausdorff dimension for bounded set

Let $F$ be the set of of numbers $x\in [0,1]$ with base 3 expansions $0.a_1a_2...$ for which there exists an integer k such that $a_i\neq 1$ for all $i\geq k$. Find the Hausdorff dimension of $F$. ...
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### Hausdorff measure finite and positive

Let $A\subseteq [0,1]^d$ be a Borel measurable set. Let $d_H(A)$ be the Hausdorff dimension of $A$. Let $\mathcal{H}^{d_H(A)}$ be the Hausdorff measure w.r.t to the dimension of $A$. My question is, ...
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### Hausdorff decomposition of a measure

Let $\mu$ be a probability measure on $(\mathbb R^n,\mathcal B^n)$. Is it true that $\mu$ can be decomposed as a countable sum of measures which are only distributed on sets of a fixed Hausdorff ...
Hi there I am struggling to understand the Hausdorff dimension of the Sierpinski triangle $S$. Below is I did to prove that $\alpha=\frac{\log 3}{\log 2}$, what should I do for $\alpha \le \frac{\log ... 1answer 265 views ### What is the fractal dimension/Hausdorff dimension of a Koch's snowflake? I have found that the fractal dimension of a self-similar object is: $$\text{fractal dimension} = \frac{\log(\text{number of self-similar pieces})}{\log(\text{magnification factor})}$$ See here ... 1answer 393 views ### Hausdorff dimension of Hamiltonian orbit closure and symplectic leaves Let$\dot{x} = \Pi \cdot \nabla H$be a smooth Hamiltonian-Poisson system on$\mathbb{R}^n$.$H: \mathbb{R}^n \to \mathbb{R}$is the Hamiltonian and$\Pi = (\Pi^{ij})$is a skew-symmetric matrix of ... 0answers 70 views ### Hausdorff dimension of a dense orbit in the Lorenz attractor If I am not mistaken, then: the Lorenz attractor$\mathcal{A}$has Hausdorff dimension$\dim_H(\mathcal{A}) > 2$, and the Lorenz attractor$\mathcal{A}$contains a dense orbit$\mathcal{O}$, i.e. ... 1answer 102 views ### Hausdorff dimension of compact set Let$(X, d)$be a metric space and$S \subset X$. Let$\DeclareMathOperator{\diam}{diam}\diam(S)$denote diameter of$S$, that is$\diam(S) = \sup \{ d(x, y) \colon \: x, y \in S \}$. Let$\delta > ...
Let $A\subset [0,1]$ be a compact set of Hausdorff dimension $\alpha$. Let also $A_n:=\{x\in \mathbb{R}^n~:~ \|x\|_2 \in A\}$. Is it true that $\text{dim}_H(A_n)=n-1+\alpha$? I believe that this ...