Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 221–243 | Cite as

Counting exceptional points for rational numbers associated to the Fibonacci sequence

  • Charles L. SamuelsEmail author


If \(\alpha \) is a non-zero algebraic number, we let \(m(\alpha )\) denote the Mahler measure of the minimal polynomial of \(\alpha \) over \(\mathbb Z\). A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the t-metric Mahler measure, denoted \(m_t(\alpha )\). For fixed \(\alpha \in \overline{\mathbb Q}\), the map \(t\mapsto m_t(\alpha )\) is continuous, and moreover, is infinitely differentiable at all but finitely many points, called exceptional points for \(\alpha \). It remains open to determine whether there is a sequence of elements \(\alpha _n\in \overline{\mathbb Q}\) such that the number of exceptional points for \(\alpha _n\) tends to \(\infty \) as \(n\rightarrow \infty \). We utilize a connection with the Fibonacci sequence to formulate a conjecture on the t-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.


Mahler measure Metric Mahler measure Fibonacci numbers 


  1. 1.
    D. Bertsimas, J.N. Tsitsiklis, Introduction to Linear Optimization, 3rd edn. (Athena Scientific, Nashua). ISBN-13: 978-1886529199 (1997)Google Scholar
  2. 2.
    P. Borwein, E. Dobrowolski, M.J. Mossinghoff, Lehmer’s problem for polynomials with odd coefficients. Ann. Math. 166(2), 347–366 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    P. Chebyshev, Mémoire sur les nombres premiers. J. Math. Pures Appl. 1, 366–390 (1852)Google Scholar
  4. 4.
    E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34(4), 391–401 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Dubickas, C.J. Smyth, On the metric Mahler measure. J. Number Theory 86, 368–387 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    A. Dubickas, C.J. Smyth, On metric heights. Period. Math. Hung. 46(2), 135–155 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    J. Jankauskas, C.L. Samuels, The \(t\)-metric Mahler measures of surds and rational numbers. Acta Math. Hung. 134(4), 481–498 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    L. Kronecker, Näherungsweise ganzzahlige Auflösung linearer Gleichungen, Berl. Ber. 1179–1193 and 1271–1299 (1884)Google Scholar
  9. 9.
    D.H. Lehmer, Factorization of certain cyclotomic functions. Ann. Math. 34, 461–479 (1933)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    M.J. Mossinghoff, Lehmer’s Problem.
  11. 11.
    C.L. Samuels, The infimum in the metric Mahler measure. Can. Math. Bull. 54, 739–747 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    C.L. Samuels, A collection of metric Mahler measures. J. Ramanujan Math. Soc. 25(4), 433–456 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    C.L. Samuels, The parametrized family of metric Mahler measures. J. Number Theory 131(6), 1070–1088 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    C.L. Samuels, Metric heights on an Abelian group. Rocky Mt. J. Math. 44(6), 2075–2091 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    C.L. Samuels, Continued fraction expansions in connection with the metric Mahler measure. Monatsh. Math. 181(4), 907–935 (2016)Google Scholar
  16. 16.
    A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number. Acta Arith. 24, 385–399 (1973). Addendum, ibid. 26(3), 329–331 (1975)Google Scholar
  17. 17.
    G. Sierksma, Y. Zwols, Linear and Integer Optimization: Theory and Practice, 3rd edn. (CRC Press, Taylor & Francis Group, Boca Raton). ISBN-13: 978-1498710169 (2015)Google Scholar
  18. 18.
    C.J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer. Bull. Lond. Math. Soc. 3, 169–175 (1971)CrossRefzbMATHGoogle Scholar
  19. 19.
    P. Voutier, An effective lower bound for the height of algebraic numbers. Acta Arith. 74, 81–95 (1996)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Department of MathematicsChristopher Newport UniversityNewport NewsUSA

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