Some computations on the spectra of Pisot and Salem numbers
HTML articles powered by AMS MathViewer
- by Peter Borwein and Kevin G. Hare;
- Math. Comp. 71 (2002), 767-780
- DOI: https://doi.org/10.1090/S0025-5718-01-01336-9
- Published electronically: November 14, 2001
Abstract:
Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdős, Joó and Komornik in 1990, is the determination of $l(q)$ for Pisot numbers $q$, where \[ l(q) = \inf (|y|: y = \epsilon _0 + \epsilon _1 q^1 + \cdots + \epsilon _n q^n, \epsilon _i \in \{\pm 1, 0\}, y \neq 0).\] 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 $a(q) \neq 0$.References
- David W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), no. 144, 1244–1260. MR 491587, DOI 10.1090/S0025-5718-1978-0491587-8
- David W. Boyd, On beta expansions for Pisot numbers, Math. Comp. 65 (1996), no. 214, 841–860. MR 1325863, DOI 10.1090/S0025-5718-96-00693-X
- Y. Bugeaud, On a property of Pisot numbers and related questions, Acta Math. Hungar. 73 (1996), no. 1-2, 33–39. MR 1415918, DOI 10.1007/BF00058941
- Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
- Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest, Introduction to algorithms, The MIT Electrical Engineering and Computer Science Series, MIT Press, Cambridge, MA; McGraw-Hill Book Co., New York, 1990. MR 1066870
- P. Erdős, I. Joó, and F. J. Schnitzer, On Pisot numbers, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 39 (1996), 95–99 (1997). MR 1451448
- P. Erdős, M. Joó, and I. Joó, On a problem of Tamás Varga, Bull. Soc. Math. France 120 (1992), no. 4, 507–521 (English, with English and French summaries). MR 1194274
- P. Erdős and V. Komornik, Developments in non-integer bases, Acta Math. Hungar. 79 (1998), no. 1-2, 57–83. MR 1611948, DOI 10.1023/A:1006557705401
- 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), no. 3, 377–390 (English, with French summary). MR 1078082
- Paul Erdős, István Joó, and Vilmos Komornik, On the sequence of numbers of the form $\epsilon _0+\epsilon _1q+\cdots +\epsilon _nq^n,\ \epsilon _i\in \{0,1\}$, Acta Arith. 83 (1998), no. 3, 201–210. MR 1611185, DOI 10.4064/aa-83-3-201-210
- Adriano M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409–432. MR 137961, DOI 10.1090/S0002-9947-1962-0137961-5
- Kevin Hare, Home page, http://www.cecm.sfu.ca/~kghare/PISOT/, 2000.
- I. N. Herstein, Topics in algebra, 2nd ed., Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. MR 356988
- Vilmos Komornik, Paola Loreti, and Marco Pedicini, An approximation property of Pisot numbers, J. Number Theory 80 (2000), no. 2, 218–237. MR 1740512, DOI 10.1006/jnth.1999.2456
- Ka-Sing Lau, Dimension of a family of singular Bernoulli convolutions, J. Funct. Anal. 116 (1993), no. 2, 335–358. MR 1239075, DOI 10.1006/jfan.1993.1116
- A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
- Maurice Mignotte, Mathematics for computer algebra, Springer-Verlag, New York, 1992. Translated from the French by Catherine Mignotte. MR 1140923, DOI 10.1007/978-1-4613-9171-5
- Yuval Peres and Boris Solomyak, Approximation by polynomials with coefficients $\pm 1$, J. Number Theory 84 (2000), 185–198.
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
Bibliographic 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
- 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.
- © Copyright 2001 by the Authors
- Journal: Math. Comp. 71 (2002), 767-780
- MSC (2000): Primary 11Y60, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-01-01336-9
- MathSciNet review: 1885627