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Numbers with simply normal β-expansions

Published online by Cambridge University Press:  26 April 2018

SIMON BAKER  and
DERONG KONG
SIMON BAKER
Affiliation:
Mathematical Institute, University of Warwick, Coventry, CV4 7ALU.K. e-mail: [email protected]
DERONG KONG
Affiliation:
Mathematical Institute, University of Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands. e-mail: [email protected]
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Abstract

In [6] the first author proved that for any β ∈ (1, βKL) every x ∈ (0, 1/(β − 1)) has a simply normal β-expansion, where βKL ≈ 1.78723 is the Komornik–Loreti constant. This result is complemented by an observation made in [22], where it was shown that whenever β ∈ (βT, 2] there exists an x ∈ (0, 1/(β − 1)) with a unique β-expansion, and this expansion is not simply normal. Here βT ≈ 1.80194 is the unique zero in (1, 2] of the polynomial x3 − x2 − 2x + 1. This leaves a gap in our understanding within the interval [βKL, βT]. In this paper we fill this gap and prove that for any β ∈ (1, βT], every x ∈ (0, 1/(β − 1)) has a simply normal β-expansion. For completion, we provide a proof that for any β ∈ (1, 2), Lebesgue almost every x has a simply normal β-expansion. We also give examples of x with multiple β-expansions, none of which are simply normal.

Our proofs rely on ideas from combinatorics on words and dynamical systems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 171 - 192
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

REFERENCES

[1] Alcaraz Barrera, R. Topological and ergodic properties of symmetric sub-shifts. Discrete Contin. Dyn. Syst. 34 (2014), no. 11, 4459–4486.Google Scholar
[2] Alcaraz Barrera, R., Baker, S. and Kong, D. Entropy, topological transitivity and dimensional properties of unique q-expansions Trans. Amer. Math. Soc. (to appear).Google Scholar
[3] Allouche, J.-P. and Cosnard, M. The Komornik–Loreti constant is transcendental. Amer. Math. Monthly 107 (2000), no. 5, 448–449.Google Scholar
[4] Allouche, J.-P., Clarke, M. and Sidorov, N. Periodic unique beta-expansions: the Sharkovskiǐ ordering. Ergodic Theory Dynam. Systems 29 (2009), 1055–1074.Google Scholar
[5] Allouche, J.-P. and Shallit, J. The ubiquitous Prouhet–Thue–Morse sequence. In Ding, C., Helleseth, T. and Niederreiter, H., eds., Sequences and their applications: Proceedings of SETA '98 (Springer-Verlag, 1999), pp. 1–16.Google Scholar
[6] Baker, S. Digit frequencies and self-affine sets with non-empty interior. arXiv:1701.06773.Google Scholar
[7] Baker, S. Generalised golden ratios over integer alphabets. Integers 14 (2014), Paper No. A15.Google Scholar
[8] Baker, S. On small bases which admit countably many expansions. J. Number Theory 147 (2015), 515–532.Google Scholar
[9] Baker, S. and Sidorov, N. Expansions in non-integer bases: lower order revisited Integers 14 (2014), Paper No. A57.Google Scholar
[10] Besicovitch, A. S. On the sum of digits of real numbers represented in the dyadic system. Math. Ann. 110 (1935), no. 1, 321–330.Google Scholar
[11] Borel, E. Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo 27 (1909), 247–271.Google Scholar
[12] Dajani, K. and de Vries, M. Measures of maximal entropy for random β-expansions. J. Eur. Math. Soc. 7 (2005), no. 1, 51–68.Google Scholar
[13] Dajani, K. and de Vries, M. Invariant densities for random β-expansions. J. Eur. Math. Soc. 9 (2007), no. 1, 157–176.Google Scholar
[14] Daróczy, Z. and Kátai, I. On the structure of univoque numbers. Publ. Math. Debrecen 46 (1995), no. 3–4, 385–408.Google Scholar
[15] de Vries, M. and Komornik, V. Unique expansions of real numbers. Adv. Math. 221 (2009), no. 2, 390–427.Google Scholar
[16] de Vries, M. and Komornik, V. Expansions in non-integer bases. Combinatorics, words and symbolic dynamics, 18–58. Encyclopedia Math. Appl. 159 (Cambridge University Press, Cambridge, 2016).Google Scholar
[17] Eggleston, H. G. The fractional dimension of a set defined by decimal properties. Quart. J. Math., Oxford Ser. 20 (1949), 31–36.Google Scholar
[18] Erdős, P., Horváth, M. and Joó, I. On the uniqueness of the expansions 1 = ∑i=1∞ q −ni. Acta Math. Hungar. 58 (1991), no. 3–4, 333–342.Google Scholar
[19] Erdős, P. and Joó, I. On the number of expansions 1 = ∑q −ni. Ann. Univ. Sci. Budapest 35 (1992), 129–132.Google Scholar
[20] Erdős, P., Joó, I. and Komornik, V. Characterization of the unique expansions 1 = ∑i=1∞ q −ni and related problems. Bull. Soc. Math. France. 118 (1990), 377–390.Google Scholar
[21] Glendinning, P. and Sidorov, N. Unique representations of real numbers in non-integer bases. Math. Res. Letters 8 (2001), 535–543.Google Scholar
[22] Jordan, T., Shmerkin, P. and Solomyak, B. Multifractal structure of Bernoulli convolutions. Math. Proc. Camb. Phils. Soc. 151 (2011), no 3, 521–539.Google Scholar
[23] Komornik, V. Expansions in noninteger bases. Integers 11B (2011), Paper No. A9, 30 pp.Google Scholar
[24] Komornik, V. and Kong, D. Bases with two expansions. arXiv: 1705.00473.Google Scholar
[25] Komornik, V., Kong, D. and Li, W. Hausdorff dimension of univoque sets and Devil's staircase. Adv. Math. 305 (2017), no 10, 165–196.Google Scholar
[26] Komornik, V. and Loreti, P. Unique developments in non-integer bases. Amer. Math. Monthly 105 (1998), no. 7, 636–639.Google Scholar
[27] Komornik, V. and Loreti, P. On the topological structure of univoque sets. J. Number Theory. 122 (2007), no. 1, 157–183.Google Scholar
[28] Kong, D. and Li, W. Hausdorff dimension of unique beta expansions. Nonlinearity 28 (2015), no. 1, 187–209.Google Scholar
[29] Kopf, C. Invariant measures for piecewise linear transformations of the interval. Appl. Math. Comput. 39 (1990), no. 2, part II, 123–144.Google Scholar
[30] Li, T.-Y. and Yorke, J. A. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183–192.Google Scholar
[31] Parry, W. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960) 401–416.Google Scholar
[32] Rényi, A. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957) 477–493.Google Scholar
[33] Sidorov, N. Almost every number has a continuum of beta-expansions. Amer. Math. Monthly 110 (2003), 838–842.Google Scholar
[34] Sidorov, N. Expansions in non-integer bases: lower, middle and top orders. J. Number Theory 129 (2009), 741–754.Google Scholar