The dimension of weakly mean porous measures: a probabilistic approach

TL;DR

This paper establishes that the packing dimension of mean porous measures is strictly less than the ambient space dimension using probabilistic methods, providing explicit bounds and improving previous estimates.

Contribution

It introduces a probabilistic approach to bound the packing dimension of weakly mean porous measures and corrects and extends prior results.

Findings

Packing dimension of mean porous measures is strictly less than ambient dimension

Provides explicit asymptotically sharp bounds for small porosity

Improves previous estimates on the dimension of weakly mean porous measures

Abstract

Using probabilistic ideas, we prove that the packing dimension of a mean porous measure is strictly smaller than the dimension of the ambient space. Moreover, we give an explicit bound for the packing dimension, which is asymptotically sharp in the case of small porosity. This result was stated in [D. B. Beliaev and S. K. Smirnov, "On dimension of porous measures", Math. Ann. 323 (2002) 123-141], but the proof given there is not correct. We also give estimates on the dimension of weakly mean porous measures, which improve another result of Beliaev and Smirnov.