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Approximation orders of real numbers by \(\beta\)-expansions. (English) Zbl 1452.11097

The authors note the following investigations:
“We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their \(\beta\)-expansions with the exponential order \(\beta^{-n}\). Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under \(\beta\)-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of \(\beta\)-expansions.”
In this paper, a survey is devoted to known results related with \(\beta\)-expansions. Basic definitions and properties for \(\beta\)-expansions are given. The separate attention is given to \(n\)-th cylinders defined in terms of \(\beta\)-expansions.
One can note the following main result of this research.
Let \(\beta>1\) be a fixed number and \(\lambda\) be the Lebesgue measure on \([0, 1]\), \[ [0,1]\ni x=\sum^{\infty} _{k=1}{\frac{\varepsilon_k(x)}{\beta^k}} \] and \[ \omega_n(x)=\sum^{n} _{k=1}{\frac{\varepsilon_k(x)}{\beta^k}}. \] Theorem. Let \(\beta>1\) be a real number. Then for \(\lambda\)-almost all \(x\in [0, 1)\), \[ \lim_{n\to\infty}{\frac{1}{n}\log_{\beta}{(x-\omega_n(x))}}=-1. \] Several main results are related to the following set \[ \left\{x\in[0,1): \liminf_{n\to\infty}{\frac{1}{\phi(n)}\log_{\beta}{(x-\omega_n(x))}}=-1\right\}, \] where \(\phi\) is a positive function defined on the set of all positive integers.
All proofs are given with explanations.

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A80 Fractals
37B10 Symbolic dynamics
37E05 Dynamical systems involving maps of the interval
37A25 Ergodicity, mixing, rates of mixing
37A50 Dynamical systems and their relations with probability theory and stochastic processes
28D05 Measure-preserving transformations
60G10 Stationary stochastic processes
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References:

[1] Aaronson, J.; Nakada, H., On the mixing coefficients of piecewise monotonic maps, Isr. J. Math., 148, 1-10 (2005) · Zbl 1085.37030
[2] Adamczewski, B.; Bugeaud, Y., Dynamics for \(\beta \)-shifts and Diophantine approximation, Ergod. Theory Dyn. Syst., 27, 6, 1695-1711 (2007) · Zbl 1140.11035
[3] Ban, J-C; Li, B., The multifractal spectra for the recurrence rates of beta-transformations, J. Math. Anal. Appl., 420, 2, 1662-1679 (2014) · Zbl 1347.37031
[4] Barreira, L.; Iommi, G., Frequency of digits in the Lüroth expansion, J. Number Theory, 129, 6, 1479-1490 (2009) · Zbl 1188.37024
[5] Blanchard, F., \( \beta \)-expansions and symbolic dynamics, Theor. Comput. Sci., 65, 2, 131-141 (1989) · Zbl 0682.68081
[6] Bradley, R., Basic properties of strong mixing conditions. A survey and some open questions, Probab. Surv., 2, 107-144 (2005) · Zbl 1189.60077
[7] Bugeaud, Y.; Wang, B-W, Distribution of full cylinders and the Diophantine properties of the orbits in \(\beta \)-expansions, J. Fractal Geom., 1, 2, 221-241 (2014) · Zbl 1309.11062
[8] Buzzi, J., Specification on the interval, Trans. Am. Math. Soc., 349, 7, 2737-2754 (1997) · Zbl 0870.58017
[9] Dajani, K.; Kraaikamp, C., On approximation by Lüroth series, J. Théor. Nombres Bordeaux, 8, 2, 331-346 (1996) · Zbl 0870.11039
[10] Falconer, K., Fractal Geometry: Mathematical Foundations and Applications (1990), Chichester: Wiley, Chichester · Zbl 0689.28003
[11] Fan, A-H; Wu, J., Approximation orders of formal Laurent series by Oppenheim rational functions, J. Approx. Theory, 121, 2, 269-286 (2003) · Zbl 1016.41010
[12] Fan, A-H; Wu, J., Metric properties and exceptional sets of the Oppenheim expansions over the field of Laurent series, Constr. Approx., 20, 4, 465-495 (2004) · Zbl 1069.11033
[13] Fan, A.; Liao, L.; Wang, B.; Wu, J., On Khintchine exponents and Lyapunov exponents of continued fractions, Ergod. Theory Dyn. Syst., 29, 1, 73-109 (2009) · Zbl 1158.37019
[14] Fan, A-H; Wang, B-W, On the lengths of basic intervals in beta expansions, Nonlinearity, 25, 5, 1329-1343 (2012) · Zbl 1256.11044
[15] Fang, L.; Wu, M.; Li, B., Limit theorems related to beta-expansion and continued fraction expansion, J. Number Theory, 163, 385-405 (2016) · Zbl 1408.11076
[16] Fang, L.; Wu, M.; Li, B., Beta-expansion and continued fraction expansion of real numbers, Acta Arith., 187, 3, 233-253 (2019) · Zbl 1448.11150
[17] Frougny, C.; Solomyak, B., Finite beta-expansions, Ergod. Theory Dyn. Syst., 12, 4, 713-723 (1992) · Zbl 0814.68065
[18] Gel’fond, A., A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat., 23, 6, 809-814 (1959) · Zbl 0092.27702
[19] Hill, R.; Velani, S., The ergodic theory of shrinking targets, Invent. Math., 119, 1, 175-198 (1995) · Zbl 0834.28009
[20] Hofbauer, F., \( \beta \)-shifts have unique maximal measure, Monatsh. Math., 85, 3, 189-198 (1978) · Zbl 0384.28009
[21] Kesseböhmer, M.; Stratmann, B., A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math., 605, 133-163 (2007) · Zbl 1117.37003
[22] Kim, D-H, The shrinking target property of irrational rotations, Nonlinearity, 20, 7, 1637-1643 (2007) · Zbl 1116.37029
[23] Kong, D-R; Li, W-X, Hausdorff dimension of unique beta expansions, Nonlinearity, 28, 1, 187-209 (2015) · Zbl 1346.37011
[24] Li, B.; Chen, Y-C, Chaotic and topological properties of \(\beta \)-transformations, J. Math. Anal. Appl., 383, 2, 585-596 (2011) · Zbl 1223.37017
[25] Li, B.; Wu, J., Beta-expansion and continued fraction expansion, J. Math. Anal. Appl., 339, 2, 1322-1331 (2008) · Zbl 1137.11053
[26] Li, B.; Persson, T.; Wang, B-W; Wu, J., Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions, Math. Z., 276, 3-4, 799-827 (2014) · Zbl 1316.11069
[27] Li, B.; Wang, B-W; Wu, J.; Xu, J., The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc. (3), 108, 1, 159-186 (2014) · Zbl 1286.11123
[28] Liao, L-M; Seuret, S., Diophantine approximation by orbits of expanding Markov maps, Ergod. Theory Dyn. Syst., 33, 2, 585-608 (2013) · Zbl 1296.37011
[29] Lü, F.; Wu, J., Diophantine analysis in beta-dynamical systems and Hausdorff dimensions, Adv. Math., 290, 919-937 (2016) · Zbl 1332.11075
[30] Parry, W., On the \(\beta \)-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11, 401-416 (1960) · Zbl 0099.28103
[31] Pfister, C.; Sullivan, W., Large deviations estimates for dynamical systems without the specification property. Applications to the \(\beta \)-shifts, Nonlinearity, 18, 1, 237-261 (2005) · Zbl 1069.60029
[32] Philipp, W., Some metrical theorems in number theory, Pac. J. Math., 20, 109-127 (1967) · Zbl 0144.04201
[33] Pollicott, M.; Weiss, H., Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation, Commun. Math. Phys., 207, 1, 145-171 (1999) · Zbl 0960.37008
[34] Rényi, A., Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8, 477-493 (1957) · Zbl 0079.08901
[35] Schmeling, J., Symbolic dynamics for \(\beta \)-shifts and self-normal numbers, Ergod. Theory Dyn. Syst., 17, 3, 675-694 (1997) · Zbl 0908.58017
[36] Schmidt, K., On periodic expansions of Pisot numbers and Salem numbers, Bull. Lond. Math. Soc., 12, 4, 269-278 (1980) · Zbl 0494.10040
[37] Seuret, S.; Wang, B-W, Quantitative recurrence properties in conformal iterated function systems, Adv. Math., 280, 472-505 (2015) · Zbl 1319.11050
[38] Tan, B.; Wang, B-W, Quantitative recurrence properties for beta-dynamical system, Adv. Math., 228, 4, 2071-2097 (2011) · Zbl 1284.11113
[39] Thompson, D., Irregular sets, the \(\beta \)-transformation and the almost specification property, Trans. Am. Math. Soc., 364, 10, 5395-5414 (2012) · Zbl 1300.37017
[40] Tong, X.; Yu, Y-L; Zhao, Y-F, On the maximal length of consecutive zero digits of \(\beta \)-expansions, Int. J. Number Theory, 12, 3, 625-633 (2016) · Zbl 1337.11053
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