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A lower bound for Garsia’s entropy for certain Bernoulli convolutions

Part of: Algebraic number theory: global fields Real functions

Published online by Cambridge University Press:  01 May 2010

Kevin G. Hare  and
Nikita Sidorov
Kevin G. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (email: kghare@math.uwaterloo.ca)
Nikita Sidorov
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom (email: sidorov@manchester.ac.uk)

Abstract

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Let β∈(1,2) be a Pisot number and let Hβ denote Garsia’s entropy for the Bernoulli convolution associated with β. Garsia, in 1963, showed that Hβ<1 for any Pisot β. For the Pisot numbers which satisfy xm=xm−1+xm−2+⋯+x+1 (with m≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier for m=2 and later by Grabner, Kirschenhofer and Tichy for m≥3, and it proves to be close to 1. No other numerical values for Hβ are known. In the present paper we show that Hβ>0.81 for all Pisot β, and improve this lower bound for certain ranges of β. Our method is computational in nature.

MSC classification

Secondary: 26A30: Singular functions, Cantor functions, functions with other special properties 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 130 - 143
Copyright
Copyright © London Mathematical Society 2010

References

[1] Alexander, J. C. and Zagier, D., ‘The entropy of a certain infinitely convolved Bernoulli measure’, J. Lond. Math. Soc. (2) 44 (1991) 121–134.CrossRefGoogle Scholar
[2] Amara, M., ‘Ensembles fermés de nombres algébriques’, Ann. Sci. Éc. Norm. Supér. (3) 83 (1966) 215–270.CrossRefGoogle Scholar
[3] Boyd, D. W., ‘Pisot numbers in the neighborhood of a limit point. II’, Math. Comp. 43 (1984) 593–602.CrossRefGoogle Scholar
[4] Boyd, D. W., ‘Pisot numbers in the neighbourhood of a limit point. I’, J. Number Theory 21 (1985) 17–43.CrossRefGoogle Scholar
[5] Erdős, P., ‘On a family of symmetric Bernoulli convolutions’, Amer. J. Math. 61 (1939) 974–976.CrossRefGoogle Scholar
[6] Feng, D.-J., ‘The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers’, Adv. Math. 195 (2005) 24–101.CrossRefGoogle Scholar
[7] Feng, D.-J. and Sidorov, N., ‘Growth rate for beta-expansions’, Preprint, http://arxiv.org/abs/0902.0488, Monatsh. Math., to appear.Google Scholar
[8] Garsia, A. M., ‘Arithmetic properties of Bernoulli convolutions’, Trans. Amer. Math. Soc. 102 (1962) 409–432.CrossRefGoogle Scholar
[9] Garsia, A. M., ‘Entropy and singularity of infinite convolutions’, Pacific J. Math. 13 (1963) 1159–1169.CrossRefGoogle Scholar
[10] Grabner, P. J., Kirschenhofer, P. and Tichy, R. F., ‘Combinatorial and arithmetical properties of linear numeration systems’, Combinatorica 22 (2002) 245–267 (Special issue: Paul Erdős and his mathematics).CrossRefGoogle Scholar
[11] Hare, K. G., ‘Home page’, http://www.math.uwaterloo.ca/∼kghare.Google Scholar
[12] Jessen, B. and Wintner, A., ‘Distribution functions and the Riemann zeta function’, Trans. Amer. Math. Soc. 38 (1935) 48–88.CrossRefGoogle Scholar
[13] Komatsu, T., ‘An approximation property of quadratic irrationals’, Bull. Soc. Math. France 130 (2002) 35–48.CrossRefGoogle Scholar
[14] Lalley, S. P., ‘Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution’, J. Lond. Math. Soc. (2) 57 (1998) 629–654.CrossRefGoogle Scholar
[15] Lothaire, M., Algebraic combinatorics on words (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
[16] Parry, W., ‘On the β-expansions of real numbers’, Acta Math. Acad. Sci. Hungar. 11 (1960) 401–416.CrossRefGoogle Scholar
[17] Peres, Y., Schlag, W. and Solomyak, B., ‘Sixty years of Bernoulli convolutions’, Fractal geometry and stochastics, vol. II (Greifswald/Koserow, 1998), Progress in Probability, 46 (Birkhäuser, Basel, 2000) 39–65.CrossRefGoogle Scholar
[18] Pushkarev, I. A., ‘The ideal lattices of multizigzags and the enumeration of Fibonacci partitions’, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 223 (1995) 280–312, English translation in J. Math. Sci. (New York) 87 (1997) 4157–4179.Google Scholar
[19] Rényi, A., ‘Representations for real numbers and their ergodic properties’, Acta Math. Acad. Sci. Hungar. 8 (1957) 477–493.CrossRefGoogle Scholar
[20] Salem, R., ‘A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan’, Duke Math. J. 11 (1944) 103–108.CrossRefGoogle Scholar
[21] Siegel, C. L., ‘Algebraic integers whose conjugates lie in the unit circle’, Duke Math. J. 11 (1944) 597–602.CrossRefGoogle Scholar
[22] Solomyak, B., ‘Notes on Bernoulli convolutions’, Fractal geometry and applications: a jubilee of Benoǐt Mandelbrot. Part 1, Proceedings of Symposia in Pure Mathematics, 72 (American Mathematical Society, Providence, RI, 2004) 207–230.CrossRefGoogle Scholar
[23] Talmoudi, F. L., ‘Sur les nombres de S∩[1,2[’, C. R. Acad. Sci. Paris Sér. A–B 287 (1978) A739–A741.Google Scholar