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A remark on exceptional sets in Erdös-Rényi limit theorem. (English) Zbl 1386.11089
The authors take into account the dyadic expansions of the numbers belonging to the interval \([0,1)\). Then, it is proved that the Hausdorff dimension of a certain exceptional set is equal to one.
Authors’ abstract: Let \((x_i)_{i=1}^{+\infty }\) be the digits sequence in the unique terminating dyadic expansion of \(x\in [0,1)\). The run-length function \(l_n(x)\) is defined by \[ l_n(x):=\max \left\{ j:x_{i+1}=x_{i+2}=\cdots =x_{i+j}=1\;\text{for some}\;0\leq i\leq n-j\right\} . \] Erdös and Rényi proved that \[ \lim _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=1, \text{a.e.}\;x\in [0,1). \] In this note, we show that for each pair of numbers \(\alpha ,\beta \in [0,+\infty ]\) with \(\alpha \leq \beta \), the following exceptional set \[ E_{\alpha ,\beta }=\left\{ x\in [0,1):\liminf _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\alpha ,\;\limsup _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\beta \right\} \] has Hausdorff dimension one.

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A80 Fractals
Full Text: DOI
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