On exceptional sets in Erdős-Rényi limit theorem

For x∈[0,1], the run-length function rn(x) is defined as the length of the longest run of 1's amongst the first n dyadic digits in the dyadic expansion of x. Erdős and Rényi proved that limn→∞⁡rn(x)log2⁡n=1 for Lebesgue almost all x∈[0,1]. In this paper, we study the Hausdorff dimensions of the exceptional sets in Erdős-Rényi limit theorem. Let φ:N→(0,+∞) be a monotonically increasing function satisfying limn→∞⁡nφ(n1+α)=+∞ with some 0<α≤1. We prove that the setEmaxφ={x∈[0,1]:liminfn→∞rn(x)φ(n)=0,limsupn→∞rn(x)φ(n)=+∞} has Hausdorff dimension one and is residual in [0,1].