# 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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### 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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Question: Suppose $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is a linear orthogonal map $(n<m)$. Prove that $$\mathcal{H}^s(T(A))=\mathcal{H}^s(A)$$ for all $0\le s$. $\\$ ${\color{red}{\text{What ... 0answers 23 views ### Show for a$\gamma-$holder map$\mathcal{H}^{s/\gamma}(f(A))\le MH^s(A)$Suppose$f: \mathbb{R}^n\rightarrow \mathbb{R}^m$is a$\gamma-$Holder continuous map$(0<\gamma\le1)$, that is:}$$|f(x)-f(y)|\le C |x-y|^\gamma, \ \ \ \ \text{for some }C>0, \text{ and all }x,... 0answers 13 views ### box dimension of set Let F consist of those numbers in$[0, 1]$whose decimal expansions do not contain the digit$5$. Find$dim_BF$. 0answers 29 views ### How do I calculate the Hausdorff dimension of a self-affine fractal (like the Barnsley Fern)? The fractal I am concerned with has an infinite number of self-affine copies of itself, and all scaled to different dimensions. And all but one of them are rotated too. I know this may sound way more '... 1answer 43 views ### 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$F$. ... 0answers 24 views ### How do you calculate Hausdorff-dimension of real world objects when Hausdorff-dimension needn't to be equal to other dimensions? In general Hausdorff-dimension is never larger as Minkowski / Box-counting dimension. For some sets like self-similar sets both dimension coincidence. In literature you can read that the Hausdorff-... 0answers 19 views ### Hausdorff dimension of graph of composition of functions Given two functions$f,g$, is there a reasonable bound of the Hausdorff dimension of the graph of$f\circ g$given the Hausdorff dimensions of the graphs of$f$and$g$? For example, does it hold that ... 0answers 18 views ### Are there interesting general properties held by sets with small Hausdorff dimension? Let us focus on the plane. I am wondering whether the sets with small (but non zero) Hausdorff dimension share some interesting properties. Let$\epsilon\in(0,1)$and assume that a set$F\subset \...
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, ...
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? ...