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The waiting spectra of the sets described by the quantitative waiting time indicators. (English) Zbl 1303.11089

Summary: The waiting spectra of the sets consisting of pairs of sequences with prescribed quantitative waiting time indicators are determined. More precisely, let \(\underline{R} (x,y)\) and \(\overline{R}(x,y)\) be the lower and upper quantitative waiting time indicators of \(y\) by \(x\) respectively in the symbolic space \(\Sigma_m\) (integer \(m \geq 2\)) and define the level sets \[ S_{\alpha ,\beta } = \left\{ (x,y) \in \Sigma _m^2:\underline{R} (x,y) = \alpha, \overline{R}(x,y) = \beta \right\}, \] where \(0 \leq \alpha \leq \beta \leq \infty\), it is shown that the sets \(S_{\alpha,\beta}\) are all of Hausdorff dimension 2. Besides, some further extensions of this result are also made.

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A78 Hausdorff and packing measures
28A80 Fractals
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