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Normality in Pisot numeration systems

Published online by Cambridge University Press:  25 August 2015

ADRIAN-MARIA SCHEERER
ADRIAN-MARIA SCHEERER*
Affiliation:
TU Graz, Austria email scheerer@math.tugraz.at
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Abstract

Copeland and Erdös [Note on normal numbers. Bull. Amer. Math. Soc. (N.S.)52 (1946), 857–860] showed that the concatenation of primes when written in base 10 yields a real number that is normal to base 10. We generalize this result to Pisot number bases in which all integers have finite expansion.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 664 - 672
Copyright
© Cambridge University Press, 2015 

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References

Bertrand-Mathis, A. and Volkmann, B.. On (𝜖, k)-normal words in connecting dynamical systems. Monatsh. Math. 107(4) (1989), 267–279.CrossRefGoogle Scholar
Besicovitch, A. S.. The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers. Math. Z. 39(1) (1935), 146–156.CrossRefGoogle Scholar
Billingsley, P.. Ergodic Theory and Information. John Wiley, New York, London, Sydney, 1965.Google Scholar
Borel, É.. Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Mat. Palermo (2) 27(1) (1909), 247–271.CrossRefGoogle Scholar
Champernowne, D. G.. The construction of decimals normal in the scale of ten. J. Lond. Math. Soc. s1–8(4) (1933), 254–260.CrossRefGoogle Scholar
Copeland, A. H. and Erdös, P.. Note on normal numbers. Bull. Amer. Math. Soc. (N.S.) 52 (1946), 857–860.CrossRefGoogle Scholar
Davenport, H. and Erdös, P.. Note on normal decimals. Canad. J. Math. 4 (1952), 58–63.CrossRefGoogle Scholar
Frougny, C. and Solomyak, B.. Finite beta-expansions. Ergod. Th. & Dynam. Sys. 12(4) (1992), 713–723.CrossRefGoogle Scholar
Frougny, C. and Steiner, W.. Minimal weight expansions in Pisot bases. J. Math. Cryptol. 2(4) (2008), 365–392.CrossRefGoogle Scholar
Grabner, P. J. and Prodinger, H.. Additive irreducibles in 𝛼-expansions. Publ. Math. Debrecen 80(3–4) (2012), 405–415.CrossRefGoogle Scholar
Hofbauer, F.. 𝛽-shifts have unique maximal measure. Monatsh. Math. 85(3) (1978), 189–198.CrossRefGoogle Scholar
Ito, S. and Shiokawa, I.. A construction of 𝛽-normal sequences. J. Math. Soc. Japan 27 (1975), 20–23.CrossRefGoogle Scholar
Madritsch, M. G.. Construction of normal numbers via pseudo-polynomial prime sequences. Acta Arith. 166(1) (2014), 81–100.CrossRefGoogle Scholar
Madritsch, M. G. and Mance, B.. Construction of $\unicode[STIX]{x1D707}$ -normal numbers. Preprint, 2012, available at http://arxiv.org/abs/1206.4950.Google Scholar
Madritsch, M. G. and Tichy, R. F.. Construction of normal numbers via generalized prime power sequences. J. Integer Seq. 16(2) (2013), Article 13.2.12.Google Scholar
Madritsch, M. G., Thuswaldner, J. M. and Tichy, R. F.. Normality of numbers generated by the values of entire functions. J. Number Theory 128(5) (2008), 1127–1145.CrossRefGoogle Scholar
Nakai, Y. and Shiokawa, I.. A class of normal numbers. Jpn. J. Math. (N.S.) 16(1) (1990), 17–29.CrossRefGoogle Scholar
Nakai, Y. and Shiokawa, I.. Discrepancy estimates for a class of normal numbers. Acta Arith. 62(3) (1992), 271–284.CrossRefGoogle Scholar
Nakai, Y. and Shiokawa, I.. Normality of numbers generated by the values of polynomials at primes. Acta Arith. 81(4) (1997), 345–356.CrossRefGoogle Scholar
Parry, W.. On the 𝛽-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.CrossRefGoogle Scholar
Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.CrossRefGoogle Scholar
Schiffer, J.. Discrepancy of normal numbers. Acta Arith. 47(2) (1986), 175–186.CrossRefGoogle Scholar