Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers


Authors: Peter Borwein and Kevin G. Hare
Journal: Trans. Amer. Math. Soc. 355 (2003), 4767-4779
MSC (2000): Primary 11Y60, 11Y40
DOI: https://doi.org/10.1090/S0002-9947-03-03333-6
Published electronically: July 24, 2003
MathSciNet review: 1997583
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number $q$, with minimal polynomial $p$, such that $p(0) = -1$, and where $p$ has only one real root, then there exists a $C(q)$, explicitly given here, such that:

(1)
For all $\epsilon > 0$, all but finitely many integer quadratics $P$ satisfy

\begin{displaymath}\vert P(q)\vert \geq \frac{C(q) - \epsilon}{H(P)^2}\end{displaymath}

where $H$ is the height function.
(2)
For all $\epsilon > 0$ there exists a sequence of integer quadratics $P_n(q)$ such that

\begin{displaymath}\vert P_n(q)\vert \leq \frac{C(q) + \epsilon}{H(P_n)^2}.\end{displaymath}

Furthermore, $C(q) < 1$ for all $q$ in this class of cubic Pisot numbers. What is surprising about this result is how precise it is, giving exact upper and lower bounds for these approximations.


References [Enhancements On Off] (What's this?)

  • 1. J. M. Borwein and P. B. Borwein, Pi and the AGM, John Wiley & Sons Inc., New York, 1998, A study in analytic number theory and computational complexity, Reprint of the 1987 original, A Wiley-Interscience Publication. MR 99h:11147
  • 2. Peter Borwein and Kevin G. Hare, Some computations on the spectra of Pisot and Salem numbers, Math. Comp. 71 (2002), no. 238, 767-780 (electronic). MR 2003a:11135
  • 3. -, General forms for minimal spectral values for a class of quadratic Pisot numbers, Bull. London Math. Soc. 35 (2003), no. 1, 47-54. MR 2003i:11154
  • 4. David W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), no. 144, 1244-1260. MR 58:10812
  • 5. J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge University Press, New York, 1957, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45. MR 19:396h
  • 6. K.O. Geddes, G. Labahn, M. B. Monagan, and S. Vorketter, The maple programming guide, Springer-Verlag, New York, 1996.
  • 7. K. G. Hare, Home page, http://www.cecm.sfu.ca/$\sim$kghare, 1999.
  • 8. I. Joó and F. J. Schnitzer, On some problems concerning expansions by noninteger bases, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 133 (1996), 3-10 (1997). MR 99b:11008
  • 9. V. Komornik, P. Loreti, and M. Pedicini, An approximation property of Pisot numbers, J. Number Theory 80 (2000), no. 2, 218-237. MR 2000k:11116
  • 10. Maurice Mignotte, Mathematics for computer algebra, Springer-Verlag, New York, 1992, Translated from the French by Catherine Mignotte. MR 92i:68071
  • 11. K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20; corrigendum, 168. MR 17:242d
  • 12. Wolfgang M. Schmidt, On simultaneous approximations of two algebraic numbers by rationals, Acta Math. 119 (1967), 27-50. MR 36:6357

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11Y60, 11Y40

Retrieve articles in all journals with MSC (2000): 11Y60, 11Y40


Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@cecm.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/S0002-9947-03-03333-6
Keywords: Pisot numbers, continued fraction, quadratic approximation
Received by editor(s): March 1, 2001
Published electronically: July 24, 2003
Additional Notes: The first author was supported by MITACS and by NSERC of Canada
The research of the second author was supported by MITACS and by NSERC of Canada
Article copyright: © Copyright 2003 copyright retained by the authors

American Mathematical Society