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An analogue of Khintchine's theorem for self-conformal sets

Published online by Cambridge University Press:  03 August 2018

SIMON BAKER
SIMON BAKER*
Affiliation:
Mathematics institute, University of Warwick, Coventry, CV4 7AL. e-mail: [email protected]
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Abstract

Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFS's). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin–Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension.

Combining our results with the mass transference principle of Beresnevich and Velani [4], we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are “very well approximated”. As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals.

The ideas put forward in this paper are introduced in the general setting of iterated function systems that may contain overlaps. We believe that by viewing iterated function systems from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 3 , November 2019 , pp. 567 - 597
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†

This research was supported by EPSRC grant EP/M001903/1.

References

REFERENCES

[1] Baker, S. Approximation properties of β-expansions. Acta Arith. 168 (2015), 269–287Google Scholar
[2] Baker, S. Approximation properties of β-expansion II. Ergodic Theory Dynam. Syst. (to appear).Google Scholar
[3] Beresnevich, V., Dickinson, D. and Velani, S. Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179 (2006), no. 846, x+91 pp.Google Scholar
[4] Beresnevich, V. and Velani, S. A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164 (2006), no. 3, 971–992.Google Scholar
[5] Bugeaud, Y. Approximation by algebraic numbers. Cambridge Tracts in Mathematics (2004).Google Scholar
[6] Duffin, R. J. and Schaeffer, A. C. Khintchine's problem in metric Diophantine approximation. Duke Math. J. 8 (1941). 243–255.Google Scholar
[7] Falconer, K. Techniques in Fractal Geometry (John Wiley, 1997).Google Scholar
[8] Feng, D. J. and Hu, H. Dimension theory of iterated function systems. Comm. Pure Appl. Math. 62 (2009), no. 11, 1435–1500.Google Scholar
[9] Fishman, L. and Simmons, D. Intrinsic approximation for fractals defined by rational iterated function systems: Mahler's research suggestion Proc. Lond. Math. Soc. (3) 109 (2014), no. 1, 189–212.Google Scholar
[10] Hochman, M. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2) 180 (2014), no. 2, 773–822.Google Scholar
[11] Hoeffding, W. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963), 13–30.Google Scholar
[12] Hutchinson, J. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), no. 5, 713–747.Google Scholar
[13] Käenmäki, A. On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 2, 419–458.Google Scholar
[14] Kasch, F. and Volkmann, B. Zur Mahlerschen Vermutung über S-Zahlen. Math. Ann. 136 (1958), 442–453.Google Scholar
[15] Khintchine, A. Einige Sätze über Kettenbruche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92 (1924), 115–125.Google Scholar
[16] Koksma, J. F. Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen. Monatsh. Math. Phys. 48 (1939), 176–189.Google Scholar
[17] Levesley, J., Salp, C. and Velani, S. On a problem of K. Mahler: Diophantine approximation and Cantor sets. Math. Ann. 338 (2007), no. 1, 97–118.Google Scholar
[18] Peres, Y. and Solomyak, B. Problems on self-similar sets and self-affine sets: an update. Fractal geometry and stochastics, II (Greifswald/Koserow, 1998). volume 46 of Progr. Probab., pages 95–106 (Birkhuser, Basel, 2000).Google Scholar
[19] Ruelle, D. Repellers for real analytic maps. Ergodic Theory Dynam. Sys. 2 (1982), no. 1, 99–107.Google Scholar
[20] Sprindžuk, V. G. A proof of Mahlers conjecture on the measure of the set of S-numbers. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 379–436.Google Scholar
[21] Sprindžuk, V. G. Mahlers Problem in Metric Number Theory. (Izdat. Nauka i Tehnika, Minsk, 1967), 181 pp.Google Scholar
[22] Sprindžuk, V. G. Metric Theory of Diophantine Approximation (V. H. Winston & Sons, 1979).Google Scholar