On the Continued Fraction Expansion of a Class of Numbers

  • Damien Roy
Part of the Developments in Mathematics book series (DEVM, volume 16)


A classical result of Dirichlet asserts that, for each real number ξ and each real X ≥ 1, there exists a pair of integers (x 0 , x1) satisfying
$$ 1 \leqslant x_0 \leqslant X and \left| {x_0 \xi - x_1 } \right| \leqslant X^{ - 1} $$
(a general reference is Chapter I of [10]). If ξ is irrational, then, by letting X tend to infinity, this provides infinitely many rational numbers x 1 /x 0 with |ξ - x1/x0 x 0 - 2. By contrast, an irrational real number ξ is said to be badly approximable if there exists a constant c1 > 0 suchthat |ξ - p/q > c 1 q - 2 for each p/q ∈ or,equivalently,if ξ has bounded partial quotients in its continued fraction expansion. Thanks to H. Davenport and W. M. Schmidt, the badly approximable real numbers can also be described as those ξ ∈ ℝ \ ℚ for which the result of Dirichlet can be improved in the sense that there exists a constant c2 < 1 such that the inequalities 1 ≤ x0X and |x0ξ - x 1 |≤ c2X-1 admit a solution (x0, x1) ∈ ℤ2 for each sufficiently large X (see Theorem 1 of [2]).


Badly approximable numbers continued fractions extremal real numbers Fibonacci sequences words 

2000 Mathematics subject classification

Primary 11J70 Secondary 11J04 11J13 


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  1. 1.
    Cassels, J.W.S.: An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1957)zbMATHGoogle Scholar
  2. 2.
    Davenport, H., Schmidt, W.M.: Dirichlet’s theorem on diophantine approximation. In: Symposia Mathematica su Teoria dei Numeri, Istituto Nazionale di Alta Matematica, Rome, 1968/69. Symp. Math., 4, pp. 113–132. Academic Press, London (1970)Google Scholar
  3. 3.
    Davenport, H., Schmidt, W.M.: Approximation to real numbers by algebraic integers. Acta Arith. 15, 393–416 (1969)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Lothaire, M.: Combinatorics on Words. Encyclopedia of Mathematics and Its Applications, vol. 17. Addison-Wesley, Reading (1983)zbMATHGoogle Scholar
  5. 5.
    Lucier, B.: Binary morphisms to ultimately periodic words. arXiv: 0805.1373v1 (2008)Google Scholar
  6. 6.
    Roy, D.: Approximation simultanée d’un nombre et de son carré. C. R. Acad. Sci. Paris Ser. I 336, 1–6 (2003)zbMATHGoogle Scholar
  7. 7.
    Roy, D.: Approximation to real numbers by cubic algebraic integers I. Proc. Lond. Math. Soc. 88, 42–62 (2004)zbMATHCrossRefGoogle Scholar
  8. 8.
    Roy, D.: Diophantine approximation in small degree. In: Goren, E.Z., Kisilevsky, H. (eds.) Number Theory: Proceedings from the 7th Conference of the Canadian Number Theory Association. CRM Proceedings and Lecture Notes, vol. 36, pp. 269–285. American Mathematical Society, Providence (2004)Google Scholar
  9. 9.
    Roy, D.: On two exponents of approximation related to a real number and its square. Can. J. Math. 59, 211–224 (2007)zbMATHGoogle Scholar
  10. 10.
    Schmidt, W.M.: Diophantine Approximation. Lect. Notes Math., vol. 785. Springer, Heidelberg (1980)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Damien Roy
    • 1
  1. 1.Département de MathématiquesUniversité d’OttawaOntarioCanada

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