Search results for "Primary 28A75"

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On the Dimension of Kakeya Sets in the First Heisenberg Group

2021

We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the $\{xoy\}$-plane is a Kakeya set with Heisenberg Hausdorff dimension 3.

Mathematics - Metric GeometryMathematics - Classical Analysis and ODEsApplied MathematicsGeneral MathematicsMathematics::Classical Analysis and ODEsClassical Analysis and ODEs (math.CA)FOS: MathematicsfraktaalitCondensed Matter::Strongly Correlated ElectronsMetric Geometry (math.MG)mittateoriaPrimary 28A75 Secondary 28A78 28A80
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Sharp estimate on the inner distance in planar domains

2020

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlev\'e length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlev\'e length bound $\kappa(E) \le\pi \mathcal{H}^1(E)$ is sharp.

Pure mathematicsMathematics - Complex VariablesGeneral MathematicsBoundary (topology)accessible pointsMetric Geometry (math.MG)31A15Domain (mathematical analysis)inner distancePlanarMathematics - Metric GeometryPrimary 28A75. Secondary 31A15Bounded functionTotally disconnected spaceMetric (mathematics)FOS: Mathematics28A75Hausdorff measureComplex Variables (math.CV)Painlevé lengthMathematics
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