×

zbMATH — the first resource for mathematics

On exceptional sets in Erdős-Rényi limit theorem. (English) Zbl 1408.11077
Summary: For \(x \in [0, 1]\), the run-length function \(r_n(x)\) is defined as the length of the longest run of 1’s amongst the first \(n\) dyadic digits in the dyadic expansion of \(x\). P. Erdős and A. Rényi [ J. Anal. Math. 23, 103–111 (1970; Zbl 0225.60015)] proved that \(\lim_{n \to \infty} \frac{r_n(x)}{\log_2 n} = 1\) for Lebesgue almost all \(x \in [0, 1]\). In this paper, we study the Hausdorff dimensions of the exceptional sets in Erdős-Rényi limit theorem. Let \(\varphi : \mathbb{N} \to(0, + \infty)\) be a monotonically increasing function satisfying \(\lim_{n \to \infty} \frac{n}{\varphi(n^{1 + \alpha})} = + \infty\) with some \(0 < \alpha \leq 1\). We prove that the set \[ E_{\max}^\varphi = \left\{x \in [0, 1] : \lim \inf_{n \to \infty} \frac{r_n(x)}{\varphi(n)} = 0, \lim \sup_{n \to \infty} \frac{r_n(x)}{\varphi(n)} = + \infty \right\} \] has Hausdorff dimension one and is residual in \([0, 1]\).

MSC:
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A78 Hausdorff and packing measures
28A80 Fractals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albeverio, S.; Pratsiovytyi, M.; Torbin, G., Topological and fractal properties of subsets of real numbers which are not normal, Bull. Sci. Math., 129, 615-630, (2005) · Zbl 1088.28003
[2] Barreira, L.; Li, J. J.; Valls, C., Irregular sets are residual, Tohoku Math. J., 66, 471-489, (2014) · Zbl 1376.37032
[3] Barreira, L.; Li, J. J.; Valls, C., Irregular sets for ratios of Birkhoff averages are residual, Publ. Mat., 58, 49-62, (2014) · Zbl 1309.37018
[4] Barreira, L.; Schmeling, J., Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116, 29-70, (2000) · Zbl 0988.37029
[5] Chen, H. B.; Wen, Z. X., The fractional dimensions of intersections of the Besicovitch sets and the Erdős-Rényi sets, J. Math. Anal. Appl., 401, 29-37, (2013) · Zbl 1261.28007
[6] Erdős, P.; Rényi, A., On a new law of large numbers, J. Anal. Math., 22, 103-111, (1970) · Zbl 0225.60015
[7] Falconer, K. J., Fractal geometry—mathematical foundations and applications, (1990), John Wiley and Sons Ltd Chichester · Zbl 0689.28003
[8] Fan, A. H.; Feng, D.-J.; Wu, J., Recurrence, dimensions and entropies, J. Lond. Math. Soc., 64, 229-244, (2001)
[9] Feng, D.-J.; Wu, J., The Hausdorff dimension of recurrent sets in symbolic spaces, Nonlinearity, 14, 81-85, (2001) · Zbl 0985.37009
[10] Hyde, J.; Laschos, V.; Olsen, L.; Petrykiewicz, I.; Shaw, A., Iterated Cesàro averages, frequencies of digits and Baire category, Acta Arith., 144, 287-293, (2010) · Zbl 1226.11077
[11] D.H. Kim, B. Li, Zero-one law of Hausdorff dimensions of the recurrence sets, preprint. · Zbl 1350.28006
[12] Li, J. J.; Wu, M., The sets of divergence points of self-similar measures are residual, J. Math. Anal. Appl., 404, 429-437, (2013) · Zbl 1304.28008
[13] Li, J. J.; Wu, M., Generic property of irregular sets in systems satisfying the specification property, Discrete Contin. Dyn. Syst., 34, 635-645, (2014) · Zbl 1280.54024
[14] Li, J. J.; Wu, M., A note on the rate of returns in random walks, Arch. Math., 102, 493-500, (2014) · Zbl 1296.54034
[15] Li, J. J.; Wu, M.; Xiong, Y., Hausdorff dimensions of the divergence points of self-similar measures with the open set condition, Nonlinearity, 25, 93-105, (2012) · Zbl 1236.28007
[16] Ma, J. H.; Wen, S. Y.; Wen, Z. Y., Egoroff’s theorem and maximal run length, Monatsh. Math., 151, 287-292, (2007) · Zbl 1170.28001
[17] Olsen, L., Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137, 43-53, (2004) · Zbl 1128.11038
[18] Olsen, L.; Winter, S., Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. Lond. Math. Soc., 67, 103-122, (2003) · Zbl 1040.28014
[19] Oxtoby, J. C., Measure and category, (1996), Springer New York
[20] Peng, L., Dimension of sets of sequences defined in terms of recurrence of their prefixes, C. R. Acad. Sci. Paris, Ser. I, 343, 129-133, (2006) · Zbl 1096.37006
[21] Pesin, Y.; Pitskel, B., Topological pressure and variational principle for non-compact sets, Funct. Anal. Appl., 18, 307-318, (1984) · Zbl 0567.54027
[22] Pratsiovytyi, M.; Torbin, G., Superfractality of the set of numbers having no frequency of n-adic digits, and fractal probability distributions, Ukrainian Math. J., 47, 971-975, (1995) · Zbl 0936.28002
[23] Révész, P., Random walk in random and non-random environments, (2005), World Scientific Singapore · Zbl 1090.60001
[24] Zou, R. B., Hausdorff dimension of the maximal run-length in dyadic expansion, Czechoslovak Math. J., 61, 881-888, (2011) · Zbl 1249.11085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.