Search results for "Packing measure"

showing 2 items of 2 documents

Combinatorial proofs of two theorems of Lutz and Stull

2021

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatori…

FOS: Computer and information sciences28A80 (primary) 28A78 (secondary)General MathematicskombinatoriikkaCombinatorial proofComputational Complexity (cs.CC)01 natural sciencesCombinatoricsMathematics - Metric GeometryHausdorff and packing measures0103 physical sciencesClassical Analysis and ODEs (math.CA)FOS: Mathematics0101 mathematicsMathematicsAlgorithmic information theoryLemma (mathematics)Euclidean spacePigeonhole principle010102 general mathematicsOrthographic projectionHausdorff spaceMetric Geometry (math.MG)Projection (relational algebra)Computer Science - Computational ComplexityMathematics - Classical Analysis and ODEsfraktaalit010307 mathematical physicsmittateoria
researchProduct

Conical upper density theorems and porosity of measures

2008

Abstract We study how measures with finite lower density are distributed around ( n − m ) -planes in small balls in R n . We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.

Mathematics(all)General Mathematics010102 general mathematicsHausdorff spaceGeometryConical surfaceType (model theory)01 natural sciencesPacking measure010104 statistics & probabilityMathematics - Classical Analysis and ODEsClassical Analysis and ODEs (math.CA)FOS: MathematicsConical upper density0101 mathematicsPorosityPorosityFinite lower densityMathematicsAdvances in Mathematics
researchProduct