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

Abstract

We study the Hausdorff dimension of the difference sets between the set of typical numbers, defined via the Erdős--Rényi 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 $\text{TpcN}\setminus \text{Normal}$ and $\text{Normal}\setminus \text{TpcN}$ are $D_2({\mathbf{\Pi }}_3^0)$-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.

14 pages