Hausdorff Measures, Dyadic Approximations and Dobi\'nski Set

TL;DR

This paper investigates the Hausdorff dimension and measure of the Dobiński set, an exceptional set related to dyadic rational approximations, using advanced measure theory and Cantor set constructions.

Contribution

It introduces new methods combining the Mass Transference Principle and willow set constructions to analyze the fractal properties of the Dobiński set.

Findings

Determines the Hausdorff dimension of the Dobiński set.

Establishes bounds on the logarithmic measure of the set.

Develops a novel approach using Cantor-like willow sets.

Abstract

Dobi\'nski set $\mathcal{D}$ is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of $\mathcal{D}$ by means of the Mass Transference Principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in $\mathcal{D}$.