Measures and the Law of the Iterated Logarithm

TL;DR

This paper investigates whether certain measures are comparable to Hausdorff or packing measures in dimension d, providing negative answers in various cases using advanced probabilistic tools.

Contribution

It offers detailed comparisons between measures and generalized Hausdorff or packing measures, employing the Law of the Iterated Logarithm and L^q-spectrum estimations.

Findings

Measures are generally not comparable to Hausdorff or packing measures in dimension d.

Provides precise conditions under which measures differ from classical geometric measures.

Uses probabilistic tools to analyze measure properties in self-similar and quasi-Bernoulli cases.

Abstract

Let m be a unidimensional measure with dimension d. A natural question is to ask if the measure m is comparable with the Hausdorff measure (or the packing measure) in dimension d. We give an answer (which is in general negative) to this question in several situations (self-similar measures, quasi-Bernoulli measures). More precisely we obtain fine comparisons between the mesure m and generalized Hausdorff type (or packing type) measures. The Law of the Iterated Logarithm or estimations of the L^q-spectrum in a neighborhood of q=1 are the tools to obtain such results.