The dimension of weakly mean porous measures: a probabilistic approach (Q2888916)

Using probabilistic ideas, we prove that if \(\mu \) is a mean porous measure on \(\mathbb R^n\), then the packing dimension of \(\mu \) is strictly smaller than n. 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 [\textit{D. B. Beliaev} and \textit{S. K. Smirnov}, ``On dimension of porous measures'', Math. Ann. 323, No. 1, 123--141 (2002; Zbl 1037.28003)], 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.