Definition of logarithmic measure

Question

I am looking for a definition of logarithmic measure (and especially set of logarithmic measure zero) - as I encountered it in one textbook without explenation and it's rather difficult to find.

Thank you.

Answer 1

I don’t know how standard this term is (because it is so little used), but the notion of “logarithmic measure zero” that I’m familiar with applies to subsets of the reals and it means that the set has Hausdorff $h$-measure zero relative to the measure function [= gauge function]

$$h(t) \; = \; \frac{1}{-\log t} \; = \; \frac{1}{\log \left(\frac{1}{t}\right)}$$

(The base of the logarithm doesn’t matter.)

Equivalently, $E \subseteq {\mathbb R}$ has logarithmic measure zero if and only if for each $\varepsilon > 0$ there exists a countable [means: finite or countably infinite] collection $\{I_n\}$ of nonempty open intervals, each of length less than $1,$ such that $E \subseteq \bigcup\limits_{n=1}^{\infty}I_n$ and $\sum\limits_{n=1}^{\infty}\frac{-1}{\log L(I_n)} < \varepsilon,$ where $L(I_n)$ is the length of $I_{n}.$

Roughly speaking, the property of having logarithmic measure zero is a “logarithmically stronger” (i.e. infinitesimally stronger) smallness notion than the property of having Hausdorff dimension zero. A natural example of a set with Hausdorff dimension zero that does not have logarithmic measure zero is the set of Liouville numbers. In fact, the set of Liouville numbers cannot be written as a countable union of sets, each of which has logarithmic measure zero.

Somewhere at home I have a collection of papers that make use logarithmic measure zero sets, but I don’t know where they are right now. If I come across them at some later time, I might return to this answer and give some references. For now, see this paper and several of the papers mentioned in the mathoverflow question/answers Behaviour of power series on their circle of convergence.