Hausdorff dimension of particularly non-normal numbers in dynamical systems fulfilling the specification property

TL;DR

This paper investigates the Hausdorff dimension of particularly non-normal numbers within certain dynamical systems, revealing that this set has full Hausdorff dimension despite having measure zero.

Contribution

It demonstrates that the set of particularly non-normal numbers has full Hausdorff dimension in dynamical systems with the specification property, extending understanding of non-normal number sets.

Findings

Set of particularly non-normal numbers has full Hausdorff dimension.

Non-normal numbers set has measure zero in these systems.

Existence of numbers with mixed digit frequency behaviors.

Abstract

In this paper, we consider non-normal numbers occurring in dynamical systems fulfilling the specification property. It has been shown that in this case the set of non-normal numbers has measure zero. In the present papers we show that a smaller set, namely the set of particularly non-normal numbers, has full Hausdorff dimension. A particularly non-normal number is a number $x$ such that there exist two digits, one whose limiting frequency in $x$ exists and another one whose limiting frequency in $x$ does not exist.