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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.

MSC:
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A80 Fractals
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[1] Ban, JC; Li, B, The multifractal spectra for the recurrence rates of beta-transformations, J. Math. Anal. Appl., 420, 1662-1679, (2014) · Zbl 1347.37031
[2] Barreira, L; Saussol, B; Schmeling, J, Higher-dimensional multifractal analysis, J. Math. Pures Appl., 81, 67-91, (2002) · Zbl 1025.37019
[3] Besicovitch, AS, On the sum of digits of real numbers represented in the dyadic system, Math. Ann., 110, 321-330, (1934) · Zbl 0009.39503
[4] Eggleston, H, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser., 20, 31-36, (1949) · Zbl 0031.20801
[5] Erdös, P; Rényi, A, On a new law of large numbers, J. Anal. Math., 22, 103-111, (1970) · Zbl 0225.60015
[6] Falconer, K.J.: Fractal Geometry, Mathematical Foundations and Application. Wiley, Chichester (1990) · Zbl 0689.28003
[7] Fan, A.H., Feng, D.J., Wu, J.: Recurrence, dimension and entropy. J. Lond. Math. Soc. 64(2), 229-244 (2001) · Zbl 1011.37003
[8] Feng, DJ; Wu, J, The Hausdorff dimension of recurrent sets in symbolic spaces, Nonlinearity, 14, 81-85, (2001) · Zbl 0985.37009
[9] Kim, D.H., Li, B.: Zero-one law of Hausdorff dimensions of the recurrent sets. arXiv:1510.00495 · Zbl 1350.28006
[10] Lau, KS; Shu, L, The spectrum of poincare recurrence, Ergod. Th. Dynam. Sys., 28, 1917-1943, (2008) · Zbl 1213.37025
[11] Li, JJ; Wu, M, On exceptional sets in Erdös-Rényi limit theorem, J. Math. Anal. Appl., 436, 355-365, (2016) · Zbl 1408.11077
[12] Li, J.J., Wu, M.: On exceptional sets in Erdös-Rényi limit theorem revisited. arXiv:1511.08903 · Zbl 0985.37009
[13] Ma, JH; Wen, SY; Wen, ZY, Egoroff’s theorem and maximal run length, Monatsh. Math., 151, 287-292, (2007) · Zbl 1170.28001
[14] Peng, L, Dimension of sets of sequences defined in terms of recurrence of their prefixes, C. R. Math. Acad. Sci. Paris, 343, 129-133, (2006) · Zbl 1096.37006
[15] Wang, B.W., Wu, J.: On the maximal run-length function in continued fractions. Ann. Univ. Sci. Budapest. Sect. Comp. 34, 247-268 (2011) · Zbl 1096.37006
[16] Zou, R.B.: Hausdorff dimension of the maximal run-length in dyadic expansion. Czechoslovak Math. J. 61(136)(4), 881-888 (2011) · Zbl 1249.11085
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