# 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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### Hausdorff dimension of the zero set of the gradient of an eikonal function

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere with respect to Lebesgue measure. What is the supremal Hausdorff dimension of the set on which $f$ is ...
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### Hausdorff dimension of the zero set of $\nabla f$

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure. What is the supremal Hausdorff dimension of the set on which $f$ ...
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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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### Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?

Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$. Let $\Omega$...
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### Fractal sets and dimensions

Can we construct two sets $E$ and $F$ meeting the following criteria $\dim_H(E) = \dim_H(F) = \dim_H(E ∩ F)$ $\dim_P(E), \dim_P(F)$, and $\dim_P(E ∩ F)$ are distinct? Here $\dim_H$ denotes the ...
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### Is Kakeya conjecture open with some additional regularity condition on Kakeya map?

in the paper, ON KAKEYA MAPS WITH REGULARITY ASSUMPTIONS, the author of the paper consider the Kakeya conjecture with some regularity on Kakeya map, Kekeya conjecture[Hausdorff dimension version] If ...
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### Natural way to thicken Brownian motion to 2D?

If we have a smooth plane curve (Hausdorff dimension 1), we can thicken it by a small amount to get a 2D set (all points within distance $\epsilon$ to the curve). What if we start with the graph of a ...
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### Bounds on convergence of two orbits in the limit set of a Schottky group

Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does ...
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Let $f:R^n\to R$ be a locally Lipchitz function. Denote $H^n$ the n-dimensional Hausdorff measure. We know that for any $H^n$-measurable subset $A\subset R^n$, for $\mathcal{L}^1$-a.e. $t\in R$, $A\... 2 votes 0 answers 47 views ### (1, 2) stability and Hausdorff dimension Recall that a set$E \subset \mathbb{R}^n$is called (1,2) stable if it satisfies the following: $$W_0^{1,2}(E) = W_0^{1,2}(E^0),$$ where$E^0$is the interior of$E$and$W_0^{1,2}(E)$is defined ... 4 votes 2 answers 1k views ### A set whose Hausdorff dimension gradually changes? Can there be a set whose Hausdorff dimension gradually changes? For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, ... • 9,691 2 votes 0 answers 94 views ### Hausdorff dimension between$(1,2)$Is there a number$c \in (1,2)$for which there exist some interesting geometric property/properties which hold for every set of Hausdorff dimension in$(1,c)$and does not hold for any set of ... • 9,210 8 votes 1 answer 188 views ### 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}^...
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Let $(X,d)$ be a metric space having Hausdorff dimension $\alpha>0$ and let $0<\beta<\alpha$. Is there a metric subspace of $X$ having Hausdorff dimension $\beta$?
A dimension function is an increasing, continuous function $% f:\mathbb R_{+}\rightarrow \mathbb R_{+}$ such that $f(r)\to 0$ as $r\to 0$. Say that a dimension function $f$ is essentially sub-linear ...