On the beta expansion for Salem numbers of degree 6

Author:
David W. Boyd

Journal:
Math. Comp. **65** (1996), 861-875

MSC (1991):
Primary 11R06, 11K16; Secondary 11Y99

DOI:
https://doi.org/10.1090/S0025-5718-96-00700-4

Supplement:
Additional information related to this article.

MathSciNet review:
1333306

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Abstract: For a given , the *beta transformation* is defined for by (mod ). The number is said to be a *beta number* if the orbit is finite, hence eventually periodic. It is known that all Pisot numbers are beta numbers, and it is conjectured that this is true for Salem numbers, but this is known only for Salem numbers of degree . Here we consider some computational and heuristic evidence for the conjecture in the case of Salem numbers of degree , by considering the set of such numbers of trace at most . Although the orbit is small for the majority of these numbers, there are some examples for which the orbit size is shown to exceed and for which the possibility remains that the orbit is infinite. There are also some very large orbits which *have* been shown to be finite: an example is given for which the preperiod length is and the period length is . This is in contrast to Salem numbers of degree where the orbit size is bounded by . An heuristic probabilistic model is proposed which explains the difference between the degree- and degree- cases. The model predicts that all Salem numbers of degree and should be beta numbers but that degree- Salem numbers can have orbits which are arbitrarily large relative to the size of . Furthermore, the model predicts that a positive proportion of Salem numbers of any fixed degree will not be beta numbers. This latter prediction is not tested here.

**1.**Anne Bertrand,*Développements en base de Pisot et répartition modulo 1*, C. R. Acad. Sci. Paris Sér. A-B**285**(1977), no. 6, A419–A421 (French, with English summary). MR**0447134****2.**A. I. Borevich and I. R. Shafarevich,*Number theory*, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. MR**0195803****3.**David W. Boyd,*Salem numbers of degree four have periodic expansions*, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 57–64. MR**1024551****4.**Leopold Flatto, Jeffrey C. Lagarias, and Bjorn Poonen,*The zeta function of the beta transformation*, Ergodic Theory Dynam. Systems**14**(1994), no. 2, 237–266. MR**1279470**, https://doi.org/10.1017/S0143385700007860**5.**A. O. Gel′fond,*A common property of number systems*, Izv. Akad. Nauk SSSR. Ser. Mat.**23**(1959), 809–814 (Russian). MR**0109817****6.**Donald E. Knuth,*The art of computer programming. Vol. 1: Fundamental algorithms*, Second printing, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969. MR**0286317****7.**W. Parry,*On the 𝛽-expansions of real numbers*, Acta Math. Acad. Sci. Hungar.**11**(1960), 401–416 (English, with Russian summary). MR**0142719**, https://doi.org/10.1007/BF02020954**8.**A. Rényi,*Representations for real numbers and their ergodic properties*, Acta Math. Acad. Sci. Hungar**8**(1957), 477–493. MR**0097374**, https://doi.org/10.1007/BF02020331**9.**Raphaël Salem,*Algebraic numbers and Fourier analysis*, D. C. Heath and Co., Boston, Mass., 1963. MR**0157941****10.**Klaus Schmidt,*On periodic expansions of Pisot numbers and Salem numbers*, Bull. London Math. Soc.**12**(1980), no. 4, 269–278. MR**576976**, https://doi.org/10.1112/blms/12.4.269**11.**Boris Solomyak,*Conjugates of beta-numbers and the zero-free domain for a class of analytic functions*, Proc. London Math. Soc. (3)**68**(1994), no. 3, 477–498. MR**1262305**, https://doi.org/10.1112/plms/s3-68.3.477

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Additional Information

**David W. Boyd**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

Email:
boyd@math.ubc.ca

DOI:
https://doi.org/10.1090/S0025-5718-96-00700-4

Keywords:
Salem numbers,
beta expansions,
polynomials,
computation

Received by editor(s):
June 20, 1994

Additional Notes:
This research was supported by a grant from NSERC.

Article copyright:
© Copyright 1996
American Mathematical Society