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Hausdorff dimension of the maximal run-length in dyadic expansion

Ruibiao Zou (2011)

Czechoslovak Mathematical Journal

For any x ∈ [ 0 , 1 ) , let x = [ ϵ 1 , ϵ 2 , ⋯ , ] be its dyadic expansion. Call r n ( x ) : = max { j ≥ 1 : ϵ i + 1 = ⋯ = ϵ i + j = 1 , 0 ≤ i ≤ n - j } the n -th maximal run-length function of x . P. Erdös and A. Rényi showed that lim n → ∞ r n ( x ) / log 2 n = 1 almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than log 2 n , is quantified by their Hausdorff dimension.

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