Journal of Mathematical Sciences

, Volume 234, Issue 5, pp 595–597 | Cite as

Alternating Sums of Elements of Continued Fractions and the Minkowski Question Mark Function

  • E. P. GolubevaEmail author

The paper considers the function A(t) (0 ≤ t ≤ 1), related to the distribution of alternating sums of elements of continued fractions. The function A(t) possesses many properties similar to those of the Minkowski function ?(t). In particular, A(t) is continuous, satisfies similar functional equations, and A′(t) = 0 almost everywhere with respect to the Lebesgue measure. However, unlike ?(t), the function A(t) is not monotonically increasing. Moreover, on any subinterval of [1, 0], it has a sharp extremum.


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  1. 1.
    D. B. Zagier, “Nombre de classes et fractions continues,” Astérisque, 24–25, 81–97 (1975).MathSciNetzbMATHGoogle Scholar
  2. 2.
    V. N. Popov, “Asymptotic formula for the sum of sums of elements of the continued fractions for numbers of the form a/p,” Zap. Nauchn. Semin. LOMI, 91, 81–93 (1979).zbMATHGoogle Scholar
  3. 3.
    E. P. Golubeva, “Lengths of the periods of the continued fraction expansion of quadratic irrationalities and the class numbers of real quadratic fields,” Zap. Nauchn. Semin. LOMI, 168, 11–22 (1985).zbMATHGoogle Scholar
  4. 4.
    H. Minkowski, Gesammelte Abhandlungen, Vol. 2, B. G. Teubner (1911).Google Scholar
  5. 5.
    A. Denjoy, “Sur une function réelle de Minkowski,” J. Math. Pures Appl., 17, No. 2, 105–151 (1938).zbMATHGoogle Scholar
  6. 6.
    R. Salem, “On some singular monotonic functions which are strictly increasing,” Trans. Amer. Math. Soc., 53, No. 3, 427–439 (1943).MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. P. Golubeva, “On a plane convex curve with a large number of lattice points,” Zap. Nauchn. Semin. POMI, 357, 22–32 (2008).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Bonch-Bruevich St. Petersburg State University of TelecommunicationsSt. PetersburgRussia

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