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MiS Preprint
90/2000

A Marstrand type theorem for measures with cube density in general dimension

Andrew Lorent

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-

Received:
Jan 9, 2001
Published:
Jan 9, 2001
MSC Codes:
28A75

Related publications

inJournal
2004 Repository Open Access
Andrew Lorent

A Marstrand type theorem for measures with cube density in general dimension

In: Mathematical proceedings of the Cambridge Philosophical Society, 137 (2004) 3, pp. 657-696