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Some computations on the spectra of Pisot and Salem numbers


Authors: Peter Borwein and Kevin G. Hare
Journal: Math. Comp. 71 (2002), 767-780
MSC (2000): Primary 11Y60, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-01-01336-9
Published electronically: November 14, 2001
MathSciNet review: 1885627
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Abstract: Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos, Joó and Komornik in 1990, is the determination of $l(q)$ for Pisot numbers $q$, where

\begin{displaymath}l(q) = \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^1 + \cdots + \epsilon_n q^n, \epsilon_i \in \{\pm 1, 0\}, y \neq 0).\end{displaymath}

Although the quantity $l(q)$ is known for some Pisot numbers $q$, there has been no general method for computing $l(q)$. This paper gives such an algorithm. With this algorithm, some properties of $l(q)$ and its generalizations are investigated.

A related question concerns the analogy of $l(q)$, denoted $a(q)$, where the coefficients are restricted to $\pm 1$; in particular, for which non-Pisot numbers is $a(q)$ nonzero? This paper finds an infinite class of Salem numbers where $a(q) \neq 0$.


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  • 1. David W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), 1244-1260. MR 58:10812
  • 2. -, On beta expansions for Pisot numbers, Math. Comp. 65 (1996), 841-860. MR 96g:11090
  • 3. Y. Bugeaud, On a property of Pisot numbers and related questions, Acta Math. Hungar. 73 (1996), 33-39. MR 98c:11113
  • 4. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, no. 138, Springer Verlag, New York, 1993. MR 94i:11105
  • 5. Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest, Introduction to algorithms, MIT Press, Cambridge, MA, 1990. MR 91i:68001
  • 6. P. Erdos, I. Joó, and F. J. Schnitzer, On Pisot numbers, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 39 (1996), 95-99 (1997). MR 98d:11127
  • 7. P. Erdos, M. Joó, and I. Joó, On a problem of Tamás Varga, Bull. Soc. Math. France 120 (1992), 507-521. MR 93m:11076
  • 8. P. Erdos and V. Komornik, Developments in non-integer bases, Acta Math. Hungar. 79 (1998), 57-83. MR 99e:11132
  • 9. Pál Erdös, István Joó, and Vilmos Komornik, Characterization of the unique expansions $1=\sum^{\infty}_{i=1} q^{-n_i}$ and related problems, Bull. Soc. Math. France 118 (1990), 377-390. MR 91j:11006
  • 10. Paul Erdos, István Joó, and Vilmos Komornik, On the sequence of numbers of the form $\epsilon_0 + \epsilon_1 q+\cdots+\epsilon_n q^n, \epsilon_i \in \{0,1\}$, Acta Arith. 83 (1998), 201-210. MR 99a:11022
  • 11. Adriano M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409-432. MR 25:1409
  • 12. Kevin Hare, Home page, http://www.cecm.sfu.ca/ kghare/PISOT/, 2000.
  • 13. I.N. Herstein, Topics in algebra, second ed., John Wiley & Sons, Toronto, 1975. MR 50:9456
  • 14. Vilmos Komornik, Paola Loreti, and Marco Pedicini, An approximation property of Pisot numbers, J. Number Theory 80 (2000), 218-327. MR 2000k:11116
  • 15. Ka-Sing Lau, Dimension of a family of singular Bernoulli convolutions, J. Funct. Anal. 116 (1993), 335-358. MR 95h:28013
  • 16. A.K. Lenstra, H.W. Lenstra Jr, and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534. MR 84a:12002
  • 17. Maurice Mignotte, Mathematics for computer algebra, Springer Verlag, New York, 1991. MR 92i:68071
  • 18. Yuval Peres and Boris Solomyak, Approximation by polynomials with coefficients $\pm 1$, J. Number Theory 84 (2000), 185-198. CMP 2001:04
  • 19. R. Salem, Power series with integral coefficients, Duke Math. J. 12 (1945), 153-172. MR 6:206b

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Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@math.sfu.ca

Kevin G. Hare
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: kghare@cecm.math.sfu.ca

DOI: https://doi.org/10.1090/S0025-5718-01-01336-9
Keywords: Pisot numbers, LLL, spectrum, beta numbers
Received by editor(s): April 12, 2000
Received by editor(s) in revised form: August 8, 2000
Published electronically: November 14, 2001
Additional Notes: Research of K.G. Hare supported by MITACS and by NSERC of Canada, and P. Borwein supported by MITACS and by NSERC of Canada.
Article copyright: © Copyright 2001 by the Authors

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