On the intersections of exceptional sets in Borel’s normal number theorem and Erdös–Rényi limit theorem

Abstract

For any [Formula: see text], let [Formula: see text] be the partial summation of the first [Formula: see text] digits in the binary expansion of [Formula: see text] and [Formula: see text] be its run-length function. The classical Borel’s normal number theorem tells us that for almost all [Formula: see text], the limit of [Formula: see text] as [Formula: see text] goes to infinity is one half. On the other hand, the Erdös–Rényi limit theorem shows that [Formula: see text] increases to infinity with the logarithmic speed [Formula: see text] as [Formula: see text] for almost every [Formula: see text] in [Formula: see text]. In this paper, we are interested in the intersections of exceptional sets arising in the above two famous theorems. More precisely, for any [Formula: see text] and [Formula: see text], we completely determine the Hausdorff dimension of the following set: [Formula: see text] where [Formula: see text] and [Formula: see text] After some minor modifications, our result still holds if we replace the denominator [Formula: see text] in [Formula: see text] with any increasing function [Formula: see text] satisfying [Formula: see text] tending to [Formula: see text] and [Formula: see text]. As a result, we also obtain that the set of points for which neither the sequence [Formula: see text] nor [Formula: see text] converges has full Hausdorff dimension.