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6 questions
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What is $3/2$-dimensional Hausdorff measure of the graph of Brownian motion?
It is well-known, and well-documented, that the Hausdorff dimension of the graph of regular $1$-dimensional Brownian motion is $3/2$ (almost surely). See for example Theorem 4.29 in "Brownian ...
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5
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2
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Existence of an $\alpha$-Hölder continuous function whose graph has positive Hausdorff measure of maximal dimension
It is standard that if $f:[0,1] \rightarrow \mathbb{R}$ is $\alpha$-Hölder continuous, then its graph has Hausdorff dimension at most $2-\alpha$. My naive expectation was that "most" graphs ...
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3
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Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency
let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\} $$
I'm studying fractal geometry and ...
5
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Naïve definition of a measure on a fractal
This question was previously posted on MSE.
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use ...
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8
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How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?
Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask:
Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...
- 12.7k
7
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1
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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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