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A Marstrand type theorem for measures with cube density in general dimension
Published online by Cambridge University Press: 02 November 2004
Abstract
With a view to generalising rectifiability and density results to more general spaces we prove the following: let $H^s$ denote Hausdorff $s$ measure in $l^n_{\infty}$. Let $s\in(0,2]$. Let $S\subset l^n_{\infty}$ be a subset of positive locally finite Hausdorff $s$-measure with the property $$\lim_{r\rightarrow 0} \frac{H^s(B_r(x)\cap S)}{\alpha(s)2^{-s}r^s}=1\;\;\;\mathrm{for}\;\;H^{s}\;a.e.\;x\in S,$$ then $s$ is an integer and $S$ has a weak tangent at almost every point.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 137 , Issue 3 , November 2004 , pp. 657 - 696
- Copyright
- © 2004 Cambridge Philosophical Society
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