Hausdorff dimension of the maximal run-length in dyadic expansion

Abstract

For any x ∈ [0, 1), let x = [ɛ ₁, ɛ ₂, …,] be its dyadic expansion. Call r _n (x):= max{j ⩾ 1: ɛ _i+1 = … = ɛ _i+j = 1, 0 ⩽ i ⩽ n − j} the n-th maximal run-length function of x. P.Erdös and A.Rényi showed that $\mathop {\lim }\limits_{n \to \infty }$ r _n (x)/log₂ n = 1 almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose runlength function assumes on other possible asymptotic behaviors than log₂ n, is quantified by their Hausdorff dimension.