A generalization of the Erdős-Rényi limit theorem and the corresponding multifractal analysis
It is known that the well-known Erdős-Rényi limit theorem characterizes the limit behaviour of the run-length functions showing especially that the run-length functions for almost all numbers in the unit interval are of the logarithmic growth rate $\log_2n$.\par The aim of the present paper is to give a generalization of the classic Erdős-Rényi limit theorem through the introduction of a more general concept of the run-length function, as well as to complement the above-mentioned generalized limit theorem to the case of $m$-adic expansions, and to study the sets in which the run-length functions are endowed with different growth rates from the viewpoint of dimensional theory.\par In particular, the authors managed to give a generalization of the classic Erdős-Rényi theorem by studying the limit behaviour of the run-length function $R^p_n(x)$ of $x\in[0,1]$ in $\overline\Sigma p$, where $1\le p\le m-1$, $m\ge 2$, are integers and $\overline\Sigma p\subset\{0,1,\dots, m-1\}$ the alphabet with $p$ elements, and to prove that $$\lim_{n\to \infty}\frac{R^p_n(x)}{\log_{m/p}n}= 1,\qquad\text{for almost all }x\in [0,1],$$ \par in the sense of the Lebesgue measure.\par Furthermore, the authors manage to complement the generalized Erdős-Rényi theorem by finding that the level set in which the above limit equals to $\alpha$, for any $0\le\alpha\le+\propto$, is of full Hausdorff dimension. (It is known that for the level set $$E_m(\phi)= \Biggl\{x\in [0,1]: \lim_{n\to\infty} \frac{R^p_n(x)}{\varphi(n)}= 1\Biggr\},$$ where $\varphi$ is a positive function defined on $N$, its Hausdorff dimension is determined when the function $\varphi$ is of some particular growth rates.)
scientific article; zbMATH DE number 6943100
scientific article