## On the dimension spectrum of infinite subsystems of continued fractions

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- by Vasileios Chousionis, Dmitriy Leykekhman and Mariusz Urbański PDF
- Trans. Amer. Math. Soc.
**373**(2020), 1009-1042 Request permission

## Abstract:

In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if $E$ is any arithmetic progression, the set of primes, or the set of squares $\{n^2\}_{n \in \mathbb {N}}$, then the continued fractions whose digits lie in $E$ have full dimension spectrum, which we denote by $DS(\mathcal {CF}_E)$. Moreover we prove that if $E$ is an infinite set of consecutive powers, then the dimension spectrum $DS(\mathcal {CF}_E)$ always contains a non-trivial interval. We also show that there exists some $E \subset \mathbb {N}$ and two non-trivial intervals $I_1, I_2$, such that $DS(\mathcal {CF}_E) \cap I_1=I_1$ and $DS(\mathcal {CF}_E) \cap I_2$ is a Cantor set. On the way we employ the computational approach of Falk and Nussbaum in order to obtain rigorous effective estimates for the Hausdorff dimension of continued fractions whose entries are restricted to infinite sets.## References

- Tim Bedford,
*Hausdorff dimension and box dimension in self-similar sets*, Proceedings of the Conference: Topology and Measure, V (Binz, 1987) Wissensch. Beitr., Ernst-Moritz-Arndt Univ., Greifswald, 1988, pp. 17–26. MR**1029553** - Jean Bourgain and Alex Kontorovich,
*On Zaremba’s conjecture*, Ann. of Math. (2)**180**(2014), no. 1, 137–196. MR**3194813**, DOI 10.4007/annals.2014.180.1.3 - Richard T. Bumby,
*Hausdorff dimensions of Cantor sets*, J. Reine Angew. Math.**331**(1982), 192–206. MR**647383**, DOI 10.1515/crll.1982.331.192 - Richard T. Bumby,
*Hausdorff dimension of sets arising in number theory*, Number theory (New York, 1983–84) Lecture Notes in Math., vol. 1135, Springer, Berlin, 1985, pp. 1–8. MR**803348**, DOI 10.1007/BFb0074599 - Vasileios Chousionis, Dmitriy Leykekhman, and Mariusz Urbański,
*The dimension spectrum of conformal graph directed Markov systems*, Selecta Math. (N.S.)**25**(2019), no. 3, Paper No. 40, 74. MR**3960790**, DOI 10.1007/s00029-019-0487-6 - V. Chousionis, J. T. Tyson, and M. Urbański,
*Conformal graph directed Markov systems on Carnot groups*, Mem. Amer. Math. Soc. (to appear). - T. W. Cusick,
*Continuants with bounded digits*, Mathematika**24**(1977), no. 2, 166–172. MR**472721**, DOI 10.1112/S0025579300009050 - T. W. Cusick,
*Continuants with bounded digits. II*, Mathematika**25**(1978), no. 1, 107–109. MR**498413**, DOI 10.1112/S002557930000930X - T. W. Cusick,
*Continuants with bounded digits. III*, Monatsh. Math.**99**(1985), no. 2, 105–109. MR**781688**, DOI 10.1007/BF01304191 - Thomas W. Cusick and Mary E. Flahive,
*The Markoff and Lagrange spectra*, Mathematical Surveys and Monographs, vol. 30, American Mathematical Society, Providence, RI, 1989. MR**1010419**, DOI 10.1090/surv/030 - Pierre Dusart,
*The $k$th prime is greater than $k(\ln k+\ln \ln k-1)$ for $k\geq 2$*, Math. Comp.**68**(1999), no. 225, 411–415. MR**1620223**, DOI 10.1090/S0025-5718-99-01037-6 - Richard S. Falk and Roger D. Nussbaum,
*$C^m$ eigenfunctions of Perron-Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in $\Bbb R^1$*, J. Fractal Geom.**5**(2018), no. 3, 279–337. MR**3827801**, DOI 10.4171/JFG/62 - Richard S. Falk and Roger D. Nussbaum,
*A new approach to numerical computation of Hausdorff dimension of iterated function systems: applications to complex continued fractions*, Integral Equations Operator Theory**90**(2018), no. 5, Paper No. 61, 46. MR**3851775**, DOI 10.1007/s00020-018-2485-z - I. J. Good,
*The fractional dimensional theory of continued fractions*, Proc. Cambridge Philos. Soc.**37**(1941), 199–228. MR**4878**, DOI 10.1017/s030500410002171x - Stefan-M. Heinemann and Mariusz Urbański,
*Hausdorff dimension estimates for infinite conformal IFSs*, Nonlinearity**15**(2002), no. 3, 727–734. MR**1901102**, DOI 10.1088/0951-7715/15/3/312 - Doug Hensley,
*The Hausdorff dimensions of some continued fraction Cantor sets*, J. Number Theory**33**(1989), no. 2, 182–198. MR**1034198**, DOI 10.1016/0022-314X(89)90005-X - Douglas Hensley,
*A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets*, J. Number Theory**58**(1996), no. 1, 9–45. MR**1387719**, DOI 10.1006/jnth.1996.0058 - Doug Hensley,
*Continued fractions*, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR**2351741**, DOI 10.1142/9789812774682 - Doug Hensley,
*Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension*, Discrete Contin. Dyn. Syst.**32**(2012), no. 7, 2417–2436. MR**2900553**, DOI 10.3934/dcds.2012.32.2417 - V. Jarnik,
*Zur metrischen theorie der diophantischen approximation*, Prace Mat. Fiz.**36**(1928), 91–106. - Oliver Jenkinson,
*On the density of Hausdorff dimensions of bounded type continued fraction sets: the Texan conjecture*, Stoch. Dyn.**4**(2004), no. 1, 63–76. MR**2069367**, DOI 10.1142/S0219493704000900 - Oliver Jenkinson and Mark Pollicott,
*Computing the dimension of dynamically defined sets: $E_2$ and bounded continued fractions*, Ergodic Theory Dynam. Systems**21**(2001), no. 5, 1429–1445. MR**1855840**, DOI 10.1017/S0143385701001687 - O. Jenkinson and M. Pollicott,
*Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of $E_2$*, Adv. Math.**325**(2018), 87–115. MR**3742587**, DOI 10.1016/j.aim.2017.11.028 - O. Jenkinson and M. Pollicott,
*Rigorous dimension estimates for cantor sets arising in zaremba theory*, preprint, 2019. - Marc Kesseböhmer and Sanguo Zhu,
*Dimension sets for infinite IFSs: the Texan conjecture*, J. Number Theory**116**(2006), no. 1, 230–246. MR**2197868**, DOI 10.1016/j.jnt.2005.04.002 - R. Daniel Mauldin and Mariusz Urbański,
*Dimensions and measures in infinite iterated function systems*, Proc. London Math. Soc. (3)**73**(1996), no. 1, 105–154. MR**1387085**, DOI 10.1112/plms/s3-73.1.105 - R. Daniel Mauldin and Mariusz Urbański,
*Conformal iterated function systems with applications to the geometry of continued fractions*, Trans. Amer. Math. Soc.**351**(1999), no. 12, 4995–5025. MR**1487636**, DOI 10.1090/S0002-9947-99-02268-0 - R. Daniel Mauldin and Mariusz Urbański,
*Graph directed Markov systems*, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003. Geometry and dynamics of limit sets. MR**2003772**, DOI 10.1017/CBO9780511543050 - Curtis T. McMullen,
*Hausdorff dimension and conformal dynamics. III. Computation of dimension*, Amer. J. Math.**120**(1998), no. 4, 691–721. MR**1637951**, DOI 10.1353/ajm.1998.0031 - David Ruelle,
*Thermodynamic formalism*, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR**511655** - Siegfried M. Rump,
*Verification methods: rigorous results using floating-point arithmetic*, Acta Numer.**19**(2010), 287–449. MR**2652784**, DOI 10.1017/S096249291000005X - J. Stoer and R. Bulirsch,
*Introduction to numerical analysis*, 2nd ed., Texts in Applied Mathematics, vol. 12, Springer-Verlag, New York, 1993. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR**1295246**, DOI 10.1007/978-1-4757-2272-7 - Mariusz Urbański and Anna Zdunik,
*Continuity of the Hausdorff measure of continued fractions and countable alphabet iterated function systems*, J. Théor. Nombres Bordeaux**28**(2016), no. 1, 261–286 (English, with English and French summaries). MR**3464621**, DOI 10.5802/jtnb.938

## Additional Information

**Vasileios Chousionis**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3004
- MR Author ID: 858546
- Email: vasileios.chousionis@uconn.edu
**Dmitriy Leykekhman**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3004
- MR Author ID: 680657
- Email: dmitriy.leykekhman@uconn.edu
**Mariusz Urbański**- Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203-5017
- Email: urbanski@unt.edu
- Received by editor(s): June 11, 2018
- Received by editor(s) in revised form: May 5, 2019
- Published electronically: November 5, 2019
- Additional Notes: The first author was supported by the Simons Foundation via the project “Analysis and dynamics in Carnot groups”, Collaboration Grant no. 521845

The second author was supported by NSF grant no. DMS-1522555 - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 1009-1042 - MSC (2010): Primary 37D35, 28A80, 11K50, 11J70; Secondary 37B10, 37C30, 37C40
- DOI: https://doi.org/10.1090/tran/7984
- MathSciNet review: 4068257