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- W2622783475 abstract "Let β > 1 be a real number. For any x ∈ [ 0 , 1 ] , the run-length function r n ( x , β ) is defined as the length of the longest run of 0's amongst the first n digits in the β -expansion of x . Let { δ n } n ≥ 1 be a non-decreasing sequence of integers and define E ( { δ n } n ≥ 1 ) = { x ∈ [ 0 , 1 ] : lim sup n → ∞ r n ( x , β ) δ n = 1 } . In this paper, we show that dim H E ( { δ n } n ≥ 1 ) = max { 0 , 1 − lim inf n → ∞ δ n ⧸ n } . Using the same method, we also study a class of extremely refined subset of the exceptional set in Erdös–Rényi limit theorem. Precisely, we prove that if lim inf n → ∞ δ n n = 0 , then the set E max ( { δ n } n ≥ 1 ) = { x ∈ [ 0 , 1 ] : lim inf n → ∞ r n ( x , β ) δ n = 0 , lim sup n → ∞ r n ( x , β ) δ n = + ∞ } has full Hausdorff dimension." @default.
- W2622783475 created "2017-06-15" @default.
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- W2622783475 date "2017-11-01" @default.
- W2622783475 modified "2023-10-17" @default.
- W2622783475 title "Hausdorff dimension of some sets arising by the run-length function of β-expansions" @default.
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- W2622783475 doi "https://doi.org/10.1016/j.jmaa.2017.06.001" @default.
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