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arXiv · 2608.15866

On the Hausdorff dimension of the difference sets between typical and normal numbers

Bill Mance·Jakub Tomaszewski
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Abstract

We study the Hausdorff dimension of the difference sets between the set of typical numbers, defined via the Erd\H{o}s--R\'enyi law governing the longest run of digits, and the set of numbers normal in base $2$. Although both properties hold for Lebesgue-almost every real number, neither implies the other. The resulting difference sets $\textit{TpcN}\setminus\textit{Normal}$ and $\textit{Normal}\setminus\textit{TpcN}$ are $D_2({\bf \Pi}_3^0)$-complete in the Borel hierarchy. We show that this logical complexity is matched geometrically: both difference sets have full Hausdorff dimension. Unlike the classical Besicovitch-type sets governing single-digit frequencies, normality requires the correct frequency of every finite block simultaneously, and the explicit Moran-set constructions that succeed for simple normality break down under this requirement. We instead establish the dimension via a counting argument bypassing the need for an explicit construction.

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Bill Mance, Jakub Tomaszewski. 2026-08-16. On the Hausdorff dimension of the difference sets between typical and normal numbers. https://arxiv.org/abs/2608.15866

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