Hausdorff dimension of the maximal run-length in dyadic expansion

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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.Erdos and A.Renyi 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.