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Some comments on Garsia numbers


Authors: Kevin G. Hare and Maysum Panju
Journal: Math. Comp. 82 (2013), 1197-1221
MSC (2010): Primary 11K16, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-2012-02636-6
Published electronically: August 7, 2012
MathSciNet review: 3008855
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Abstract: A Garsia number is an algebraic integer of norm $ \pm 2$ such that all of the roots of its minimal polynomial are strictly greater than $ 1$ in absolute value. Little is known about the structure of the set of Garsia numbers. The only known limit point of positive real Garsia numbers was $ 1$ (given, for example, by the set of Garsia numbers $ 2^{1/n}$). Despite this, there was no known interval of [1,2] where the set of positive real Garsia numbers was known to be discrete and finite. The main results of this paper are:

  • An algorithm to find all (complex and real) Garsia numbers up to some fixed degree. This was performed up to degree $ 40$.
  • An algorithm to find all positive real Garsia numbers in an interval $ [c, d]$ with $ c > \sqrt {2}$.
  • There exist two isolated limit points of the positive real Garsia numbers greater than $ \sqrt {2}$. These are $ 1.618\cdots $ and $ 1.465\cdots $, the roots of $ z^2-z-1$ and $ z^3-z^2-1$, respectively. There are no other limit points greater than $ \sqrt {2}$.
  • There exist infinitely many limit points of the positive real Garsia numbers, including $ \lambda _{m,n}$, the positive real root of $ z^m -z^n - 1$, with $ m > n$.

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

  • 1. Mohamed Amara, Ensembles fermés de nombres algébriques, Ann. Sci. École Norm. Sup. (3) 83 (1966), 215-270 (1967). MR 0237459 (38:5741)
  • 2. David W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), no. 144, 1244-1260. MR 0491587 (58:10812)
  • 3. -, Pisot numbers in the neighborhood of a limit point. II, Math. Comp. 43 (1984), no. 168, 593-602. MR 758207 (87c:11096b)
  • 4. -, Pisot numbers in the neighbourhood of a limit point. I, J. Number Theory 21 (1985), no. 1, 17-43. MR 804914 (87c:11096a)
  • 5. Horst Brunotte, On Garcia numbers, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 25 (2009), no. 1, 9-16. MR 2505180 (2010b:11143)
  • 6. -, A class of quadrinomial Garsia numbers, (preprint).
  • 7. Péter Burcsi and Attila Kovács, Exhaustive search methods for CNS polynomials, Monatsh. Math. 155 (2008), no. 3-4, 421-430. MR 2461586 (2009h:11018)
  • 8. Qirong Deng, The absolute continuity of a family of self-similar measures, Int. J. Nonlinear Sci. 5 (2008), no. 2, 178-183. MR 2390971 (2009j:28018)
  • 9. Paul Erdös, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974-976. MR 0000311 (1:52a)
  • 10. A. M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409-432. MR 0137961 (25:1409)
  • 11. Kevin G. Hare, Home page, http://www.math.uwaterloo.ca/$ \sim $kghare, 2010.
  • 12. Børge Jessen and Aurel Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48-88. MR 1501802
  • 13. Attila Kovács, Generalized binary numbers systems, Ann. Univ. Sci. Budapest. Sect. Comput 20 (2001), 195-206. MR 2241084 (2007b:11006)
  • 14. James McKee and Chris Smyth, Cyclotomic polynomials with interlacing roots, manuscript.
  • 15. Yuval Peres, Wilhelm Schlag, and Boris Solomyak, Sixty years of Bernoulli convolutions, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), Progr. Probab., vol. 46, Birkhäuser, Basel, 2000, pp. 39-65. MR 1785620 (2001m:42020)
  • 16. Raphaël Salem, Algebraic numbers and Fourier analysis, D. C. Heath and Co., Boston, Mass., 1963.
  • 17. I Schur, Über potenzreihen die im inneren des einheitskreises beschrankt sind., J. Reine Angew. Math. 147 (1917), 205-232.
  • 18. -, Über potenzreihen die im inneren des einheitskreises beschrankt sind., J. Reine Angew. Math. 148 (1918), 128-145.
  • 19. C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169-175. MR 0289451 (44:6641)
  • 20. Boris Solomyak, Notes on Bernoulli convolutions, Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 207-230. MR 2112107 (2005i:26026)

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Additional Information

Kevin G. Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo Ontario, Canada, N2L 3G1
Email: kghare@math.uwaterloo.ca

Maysum Panju
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo Ontario, Canada, N2L 3G1
Email: mhpanju@rogers.com

DOI: https://doi.org/10.1090/S0025-5718-2012-02636-6
Received by editor(s): September 7, 2011
Received by editor(s) in revised form: October 13, 2011
Published electronically: August 7, 2012
Additional Notes: The first author’s research was partially supported by NSERC
The second author’s research was supported by NSERC, the UW President’s Research Award, UW Undergraduate Research Internship, and the department of Pure Mathematics at the University of Waterloo
Computational support provided by CFI/OIT grant
Article copyright: © Copyright 2012 By the authors

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