# Scaling properties of Hausdorff and packing measures

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## Abstract.

Let \(m\in{\mathbb N}\). Let \(\theta\) be a continuous increasing function defined on \({\mathbb R}^+\), for which \(\theta(0)=0\) and \(\theta(t)/t^m\) is a decreasing function of *t*. Let \(\|\cdot\|\) be a norm on \({\mathbb R}^m\), and let \(\rho\), \({\mathcal H}^\theta ={\mathcal H}^\theta_\rho\), \({\mathcal P}^\theta ={\mathcal P}^\theta_\rho\) denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions \(\theta\) such that the corresponding Hausdorff or packing measure scales with exponent \(\alpha\) by showing it must be of the form \(\theta(t) = t^\alpha L(t)\), where *L* is slowly varying. We also show that for continuous increasing functions \(\theta\) and \(\eta\) defined on \({\mathbb R}^+\), for which \(\theta(0)=\eta(0)=0\), \({\mathcal H}^\theta={\mathcal P}^\eta\) is either trivially true or false: we show that if \({\mathcal H}^\theta={\mathcal P}^\eta\), then \({\mathcal H}^\theta={\mathcal P}^\eta=c\cdot\lambda\) for a constant *c*, where \(\lambda\) is the Lebesgue measure on \({\mathbb R}^m\).

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