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A Marstrand theorem for measures with polytope density

Authors:	Andrew Lorent

Institution:	(1) Mathematical Institute, 24-29 St Giles, Oxford, UK;(2) Present address: MPI for Mathematics, Inselstrasse 22, Leipzig, Germany

Abstract:	Given ${s\in (0,2]}$ and any centrally symmetric convex polytope ${\Theta\subset\mathbb{R}^{n}}$ , define ${\Theta_r(x):=r\Theta+x}$ we prove that if a Radon measure μ has the property ${\label{df1} 0 < \mathop {\lim }\limits_{r \to 0} \frac{\mu\big(\Theta_r(x)\big)}{r^s} < \infty\quad {\rm for}\mu\textrm{ a.e. }x}$ then s is an integer. For the case Θ is the Euclidean ball, this result was first proved by Marstrand in 1955 for Hausdorff measure in the plane (Marstrand in Proc Lond Math Soc 3(4):257–302, 1954) and later for general Radon measures in ${\mathbb {R}^n}$ (Marstrand in Trans Am Math Soc 205:369–392, 1964).

Keywords:	28A75
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