Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T16:22:00.729Z Has data issue: false hasContentIssue false
  • English
  • Français

The sets of Dirichlet non-improvable numbers versus well-approximable numbers

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

Published online by Cambridge University Press:  27 June 2019

AYREENA BAKHTAWAR ,
PHILIP BOS  and
MUMTAZ HUSSAIN
AYREENA BAKHTAWAR
Affiliation:
Department of Mathematics and Statistics, La Trobe University, PO Box 199, Bendigo3552, Australia email A.Bakhtawar@latrobe.edu.au, P.Bos@atrobe.edu.au, m.hussain@latrobe.edu.au
PHILIP BOS
Affiliation:
Department of Mathematics and Statistics, La Trobe University, PO Box 199, Bendigo3552, Australia email A.Bakhtawar@latrobe.edu.au, P.Bos@atrobe.edu.au, m.hussain@latrobe.edu.au
MUMTAZ HUSSAIN
Affiliation:
Department of Mathematics and Statistics, La Trobe University, PO Box 199, Bendigo3552, Australia email A.Bakhtawar@latrobe.edu.au, P.Bos@atrobe.edu.au, m.hussain@latrobe.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$th partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$th convergent. The set of $\unicode[STIX]{x1D6F9}$-Dirichlet non-improvable numbers,

$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
is related with the classical set of $1/q^{2}\unicode[STIX]{x1D6F9}(q)$-approximable numbers ${\mathcal{K}}(\unicode[STIX]{x1D6F9})$ in the sense that ${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$. Both of these sets enjoy the same $s$-dimensional Hausdorff measure criterion for $s\in (0,1)$. We prove that the set $G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$ is uncountable by proving that its Hausdorff dimension is the same as that for the sets ${\mathcal{K}}(\unicode[STIX]{x1D6F9})$ and $G(\unicode[STIX]{x1D6F9})$. This gives an affirmative answer to a question raised by Hussain et al [Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika 64(2) (2018), 502–518].

Keywords

uniform Diophantine approximationDirichlet’s theoremJarnik theoremHausdorff dimension

MSC classification

Primary: 11K50: Metric theory of continued fractions 11K60: Diophantine approximation 11J70: Continued fractions and generalizations 11J83: Metric theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3217 - 3235
Check for updates
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beresnevich, V., Dickinson, D. and Velani, S.. Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179(846) (2006).Google Scholar
Bugeaud, Y.. Exponents of Diophantine approximation. Dynamics and Analytic Number Theory (London Mathematical Society Lecture Note Series, 437). Cambridge University Press, Cambridge, 2016, pp. 96–135.Google Scholar
Cassels, J. W. S.. An Introduction to Diophantine Approximation (Cambridge Tracts in Mathematics and Mathematical Physics, 45). Cambridge University Press, Cambridge, 1957.Google Scholar
Davenport, H. and Schmidt, W. M.. Dirichlet’s theorem on diophantine approximation. Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69). Academic Press, London, 1970, pp. 113–132.Google Scholar
Diviš, B.. A analog to the Lagrange numbers. J. Number Theory 4 (1972), 274–285.Google Scholar
Diviš, B. and Novák, B.. A remark on the theory of Diophantine approximations. Comment. Math. Univ. Carolin. 12 (1971), 127–141.Google Scholar
Dodson, M. M.. Star bodies and Diophantine approximation. J. Lond. Math. Soc. (2) 44(1) (1991), 1–8.Google Scholar
Dodson, M. M.. Hausdorff dimension, lower order and Khintchine’s theorem in metric Diophantine approximation. J. Reine Angew. Math. 432 (1992), 69–76.Google Scholar
Falconer, K.. Fractal Geometry, 3rd edn. Wiley, Chichester, 2014.Google Scholar
Haas, A.. The relative growth rate for partial quotients. New York J. Math. 14 (2008), 139–143.Google Scholar
Huang, L. and Wu, J.. Uniformly non-improvable Dirichlet set via continued fractions. Proc. Amer. Math. Soc. (2019), to appear, doi:10.1090/proc/14587.Google Scholar
Hussain, M. and Simmons, D.. A general principle for Hausdorff measure. Proc. Amer. Math. Soc., to appear, doi:10.1090/proc/14539.Google Scholar
Hussain, M., Kleinbock, D., Wadleigh, N. and Wang, B.-W.. Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika 64(2) (2018), 502–518.Google Scholar
Khintchine, A. Ya.. Continued Fractions. P. Noordhoff, Groningen, 1963, translated by Peter Wynn.Google Scholar
Kim, D. H. and Liao, L.. Dirichlet uniformly well-approximated numbers. Int. Math. Res. Not., to appear. Preprint, 2015, arXiv:1508.00520.Google Scholar
Kleinbock, D. and Wadleigh, N.. A zero-one law for improvements to Dirichlet’s theorem. Proc. Amer. Math. Soc. 146(5) (2018), 1833–1844.Google Scholar
Kristensen, S.. Metric Diophantine approximation—from continued fractions to fractals. Diophantine Analysis: Course Notes from a summer School (Trends Math.). Ed. Steuding, J.. Birkhäuser/Springer, Cham, 2016, pp. 61–127.Google Scholar
Legendre, A.-M.. Essai sur la théorie des nombres (Essay on Number Theory) (Cambridge Library Collection). Reprint of the second (1808) edn. Cambridge University Press, Cambridge, 2009, (French).Google Scholar
Daniel Mauldin, R. and Urbański, M.. Conformal iterated function systems with applications to the geometry of continued fractions. Trans. Amer. Math. Soc. 351(12) (1999), 4995–5025.Google Scholar
Waldschmidt, M.. Recent Advances in Diophantine Approximation (Number Theory, Analysis and Geometry). Springer, New York, 2012, pp. 659–704.Google Scholar
Wang, B.-W. and Wu, J.. Hausdorff dimension of certain sets arising in continued fraction expansions. Adv. Math. 218(5) (2008), 1319–1339.Google Scholar