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A DICHOTOMY LAW FOR THE DIOPHANTINE PROPERTIES IN $\unicode[STIX]{x1D6FD}$ -DYNAMICAL SYSTEMS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  16 May 2016

Michael Coons ,
Mumtaz Hussain  and
Bao-Wei Wang
Michael Coons
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia email michael.coons@newcastle.edu.au
Mumtaz Hussain
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia email mumtaz.hussain@newcastle.edu.au, drhussainmumtaz@gmail.com
Bao-Wei Wang
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China email bwei_wang@hust.edu.cn
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Abstract

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\,\text{mod}\,1$ . Further, define

$$\begin{eqnarray}W_{y}(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{x\in [0,1]:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\}\end{eqnarray}$$
and
$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\},\end{eqnarray}$$
 where $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{{>}0}$ is a positive function such that $\unicode[STIX]{x1D6F9}(n)\rightarrow 0$ as $n\rightarrow \infty$ . In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalized Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11K60: Diophantine approximation 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 884 - 897
Copyright
Copyright © University College London 2016 

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References

Badziahin, D., Beresnevich, V. and Velani, S., Inhomogeneous theory of dual Diophantine approximation on manifolds. Adv. Math. 232 2013, 1–35; MR 2989975.CrossRefGoogle Scholar
Badziahin, D., Harrap, S. and Hussain, M., An inhomogeneous Jarník type theorem for planar curves. Preprint, 2015, arXiv:1503.04981.CrossRefGoogle Scholar
Baker, A. and Schmidt, W., Diophantine approximation and Hausdorff dimension. Proc. Lond. Math. Soc. (3) 21 1970, 1–11; MR 0271033 (42 #5916).CrossRefGoogle Scholar
Beresnevich, V., Dickinson, D. and Velani, S., Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179(846) 2006; MR 2184760 (2007d:11086).Google Scholar
Beresnevich, V. and Velani, S., A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) 2006, 971–992; MR 2259250 (2008a:11090).CrossRefGoogle Scholar
Beresnevich, V. and Velani, S., Schmidt’s theorem, Hausdorff measures, and slicing. Int. Math. Res. Not. IMRN 2006, 24, Art. ID 48794; MR 2264714 (2007h:11090).Google Scholar
Bernik, V. I. and Dodson, M. M., Metric Diophantine Approximation on Manifolds (Cambridge Tracts in Mathematics 137 ), Cambridge University Press (Cambridge, 1999); MR 1727177 (2001h:11091).CrossRefGoogle Scholar
Bugeaud, Y., On the 𝛽-expansion of an algebraic number in an algebraic base 𝛽. Integers 9(A20) 2009, 215–226; MR 2534910 (2010i:11101).CrossRefGoogle Scholar
Bugeaud, Y. and Wang, B., Distribution of full cylinders and the Diophantine properties of the orbits in 𝛽-expansions. J. Fractal Geom. 1(2) 2014, 221–241; MR 3230505.CrossRefGoogle Scholar
Chernov, N. and Kleinbock, D., Dynamical Borel–Cantelli lemmas for Gibbs measures. Israel J. Math. 122 2001, 1–27; MR 1826488 (2002h:37003).CrossRefGoogle Scholar
Dodson, M. M., A note on metric inhomogeneous Diophantine approximation. J. Aust. Math. Soc. Ser. A 62(2) 1997, 175–185; MR 1433207 (98b:11085).CrossRefGoogle Scholar
Dodson, M. M., Rynne, B. P. and Vickers, J. A. G., Diophantine approximation and a lower bound for Hausdorff dimension. Mathematika 37(1) 1990, 59–73; MR 1067887 (91i:11098).CrossRefGoogle Scholar
Falconer, K., Fractal Geometry, 2nd edn (Mathematical Foundations and Applications), John Wiley & Sons, Inc. (Hoboken, NJ, 2003); MR 2118797 (2006b:28001).CrossRefGoogle Scholar
Fan, A.-H. and Wang, B., On the lengths of basic intervals in beta expansions. Nonlinearity 25(5) 2012, 1329–1343; MR 2914142.CrossRefGoogle Scholar
Fuchs, M. and Kim, D., On Kurzweil’s 0-1 law in inhomogeneous diophantine approximation. Acta. Arith. 173 2016, 41–57, doi:10.4064/aa8219-1-2016.Google Scholar
Ge, Y. and Lü, F., A note on inhomogeneous Diophantine approximation in beta-dynamical system. Bull. Aust. Math. Soc. 91(1) 2015, 34–40; MR 3294256.CrossRefGoogle Scholar
Harman, G., Metric Number Theory (London Mathematical Society Monographs. New Series 18 ), The Clarendon Press, Oxford University Press (New York, 1998); MR 1672558 (99k:11112).CrossRefGoogle Scholar
Hill, R. and Velani, S., The ergodic theory of shrinking targets. Invent. Math. 119(1) 1995, 175–198; MR 1309976 (96e:58088).CrossRefGoogle Scholar
Hill, R. and Velani, S., The shrinking target problem for matrix transformations of tori. J. Lond. Math. Soc. (2) 60(2) 1999, 381–398; MR 1724857 (2000i:37003).CrossRefGoogle Scholar
Hill, R. and Velani, S., A zero-infinity law for well-approximable points in Julia sets. Ergodic Theory Dynam. Systems 22(6) 2002, 1773–1782; MR 1944403 (2003m:37065).CrossRefGoogle Scholar
Hofbauer, F., 𝛽-shifts have unique maximal measure. Monatsh. Math. 85(3) 1978, 189–198; MR 0492180 (58 #11326).CrossRefGoogle Scholar
Hussain, M. and Kristensen, S., Metrical results on systems of small linear forms. Int. J. Number Theory 9(3) 2013, 769–782; MR 3043613.CrossRefGoogle Scholar
Hussain, M. and Kristensen, S., Metrical theorems on systems of small inhomogeneous linear forms, Preprint, 2015, arXiv:1406.3930.Google Scholar
Hussain, M. and Levesley, J., The metrical theory of simultaneously small linear forms. Funct. Approx. Comment. Math. 48(2) 2013, 167–181; MR 3100138.CrossRefGoogle Scholar
Li, B., Wang, B., Wu, J. and Xu, J., The shrinking target problem in the dynamical system of continued fractions. Proc. Lond. Math. Soc. (3) 108(1) 2014, 159–186; MR 3162824.CrossRefGoogle Scholar
Parry, W., On the 𝛽-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 1960, 401–416; MR 0142719 (26 #288).CrossRefGoogle Scholar
Persson, T. and Schmeling, J., Dyadic Diophantine approximation and Katok’s horseshoe approximation. Acta Arith. 132(3) 2008, 205–230; MR 2403650 (2009c:11111).CrossRefGoogle Scholar
Philipp, W., Some metrical theorems in number theory. Pacific J. Math. 20 1967, 109–127; MR 0205930 (34 #5755).CrossRefGoogle Scholar
Reeve, H., Shrinking targets for countable Markov maps. Preprint, 2011, arXiv:1107.4736.Google Scholar
Rényi, A., Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar 8 1957, 477–493; MR 0097374 (20 #3843).CrossRefGoogle Scholar
Schmeling, J., Symbolic dynamics for 𝛽-shifts and self-normal numbers. Ergodic Theory Dynam. Systems 17(3) 1997, 675–694; MR 1452189 (98c:11080).CrossRefGoogle Scholar
Shen, L. and Wang, B., Shrinking target problems for beta-dynamical system. Sci. China Math. 56(1) 2013, 91–104; MR 3016585.CrossRefGoogle Scholar
Tan, B. and Wang, B., Quantitative recurrence properties for beta-dynamical system. Adv. Math. 228(4) 2011, 2071–2097; MR 2836114.CrossRefGoogle Scholar
Tseng, J., On circle rotations and the shrinking target properties. Discrete Contin. Dyn. Syst. 20(4) 2008, 1111–1122; MR 2379490 (2010a:37080).CrossRefGoogle Scholar
Urbański, M., Diophantine analysis of conformal iterated function systems. Monatsh. Math. 137(4) 2002, 325–340; MR 1947918 (2004j:37085).CrossRefGoogle Scholar