Hausdorff dimension of the maximal run-length in dyadic expansion
Hausdorff dimension of the maximal run-length in dyadic expansion
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DOI：
10.1007/s10587-011-0055-5
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发表时间：
2011-12
影响因子：
0.5
通讯作者：
Ruibiao Zou
中科院分区：
文献类型：
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作者：
Ruibiao Zou
对于任意x ∈ [0，1），设x = [1，1，2，.，]是它的并元展开式. Callrn（x）：= max{j <$1：<$i+1=...=<$i+j= 1，0 <$i <$n−j} x的第n个最大游程函数。P.Erdös和A.Rényi证明了rn（x）/log 2n = 1几乎必然。本文着重分析了违反上述规律的几点。点集的大小，其游程函数假设在其他可能的渐近行为比log 2n，量化的Hausdorff维数。
For anyx∈ [0, 1), letx= [ɛ1,ɛ2, …,] be its dyadic expansion. Callrn(x):= max{j⩾ 1:ɛi+1= … =ɛi+j= 1, 0 ⩽i⩽n−j} then-th maximal run-length function ofx. P.Erdös and A.Rényi showed thatrn(x)/log2n= 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 log2n, is quantified by their Hausdorff dimension.