Continued $\mathbf{A_2}$-fractions and singular functions

  • M.V. Pratsiovytyi Institute of Mathematics of NASU, National Pedagogical Dragomanov University, Kyiv,
  • Ya. V. Goncharenko Institute of Mathematics of NASU, National Pedagogical Dragomanov University, Kyiv,
  • I.M. Lysenko Institute of Mathematics of NASU, National Pedagogical Dragomanov University,
  • S.P. Ratushniak Institute of Mathematics of NASU, National Pedagogical Dragomanov University, Kyiv,
Keywords: $A_2$-continued fraction, $A$-representation of numbers, cylinder, basic metric relation, convergent, normal property of number, singular function, cylindrical derivative

Abstract

In the article we deepen the metric component of theory of infinite $A_2$-continued fractions $[0;a_1,a_2,...,a_n,...]$ with a two-element alphabet $A_2=\{\frac12,1\}$, $a_n\in A_2$ and establish the normal property of numbers of the segment $I=[\frac12;1]$ in terms of their $A_2$-representations: $x=[0;a_1,a_2,...,a_n,...]$. It is proved that almost all (in the sense of the Lebesgue measure) numbers of segment $I$ in their $A_2$-representations use each of the tuples of elements of the alphabet of arbitrary length as consecutive digits of the representation infinitely many times. This normal property of the number is effectively used to prove the singularity of the function $f(x=[0;a_1,a_2,...,a_n,...])=e^{\sum\limits_{n=1}^{\infty}(2a_n-1)v_n},$
where $v_1+v_2+...+v_n+...$ is a given absolutely convergent series, when function $f$ is continuous (which is the case only if $v_n=\frac{v_1(-1)^{n-1}}{2^{n-1}}$, $v_1\in R$).

Author Biographies

M.V. Pratsiovytyi, Institute of Mathematics of NASU, National Pedagogical Dragomanov University, Kyiv,

Institute of Mathematics of NASU, National Pedagogical Dragomanov University
Kyiv, Ukraine

Ya. V. Goncharenko, Institute of Mathematics of NASU, National Pedagogical Dragomanov University, Kyiv,

Institute of Mathematics of NASU, National Pedagogical Dragomanov University
Kyiv, Ukraine

I.M. Lysenko, Institute of Mathematics of NASU, National Pedagogical Dragomanov University,

Institute of Mathematics of NASU, National Pedagogical Dragomanov University

S.P. Ratushniak, Institute of Mathematics of NASU, National Pedagogical Dragomanov University, Kyiv,

Institute of Mathematics of NASU, National Pedagogical Dragomanov University, Kyiv,

References

S. Albeverio, Y. Kulyba, M. Pratsiovytyi, G. Torbin, On singularity and fine spectral structure of random continued fractions, Math. Nachr., 288 (2015), 1803–1813.

A. Denjoy, Sur une fonction reelle de Minkowski, J.Math. Pures Appl., 17, (1938), 105–151.

S.O. Dmytrenko, D.V. Kyurchev, M.V. Prats’ovytyi, $A_2$-continued fraction representation of real numbers and its geometry, Ukr. Math. J., 61 (2009), №4, 541–555.

B. Jessen, A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc., 38, №1, 48–88 (1935).

E. Lukacs, Characteristic functions, Second ed., London: Griffin, 1970.

H. Minkowski, Gesammelte abhandlungen, Vol.2, Berlin, 1911, 774–794.

M.V. Pratsiovytyi, Ya.V. Goncharenko, S.O. Dmytrenko, I.M. Lysenko, S.P. Ratushniak, About one class of function with fractal properties, Bukovynian Math. J., 9, №1, 273–283 (2021).

M.V. Pratsiovytyi, Ya.V. Goncharenko, I.M. Lysenko, S.P. Ratushniak, Fractal functions of exponential type that is generated by the Q∗2 -representation of argument, Mat. Stud., 56, №2, 133–143 (2021).doi: 10.30970/ms.56.2.133-143

M. V. Pratsiovytyi, Fractal approach to investigation of singular probability distributions, National Pedagogical University, Kyiv, 1998.

M.V. Pratsiovytyi, Singularity of distributions of random variables given by distributions of elements of its continued fraction representation, Ukr. Math. J., 48, №8, 1086–1095 (1996).

M.V. Pratsiovytyi, Nowhere monotonic singular functions, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Dragomanova, Ser. 1. Fiz.-Mat. Nauky, №12, 24–36 (2011).

M.V. Pratsiovytyi, A.V. Kalashnikov, V.K. Bezborodov, Singularity of functions of a one-parameter class containing the Minkowski function, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Dragomanova, Ser. 1. Fiz.-Mat. Nauky, (2010), №11, 225–231.

M.V. Pratsiovytyi, O.V. Svynchuk, Singular non-monotone functions defined in terms of $Q_2^*$-representations of the argument, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Dragomanova, Ser. 1. Fiz.-Mat. Nauky, (2013), №15, 144–155.

U.K. Shukla, On points of non-symmetrical differentiability of continuous functions. III, Ganita, 8 (1957), 81–107.

R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 423–439.

W. Sierpi´nski, An elementary example of an increasing function that has a derivative equal to zero almost everywhere, Matematycheskyi Sbornyk, bf 30 (1916), №3, 449–473.

B.Kh. Sendov, Binary self-similar fractal functions, Fundamentalnaya i prikladnaya matematika, 5 (1999), №2, 589–595.

Published
2022-10-31
How to Cite
Pratsiovytyi, M., Goncharenko, Y. V., Lysenko, I., & Ratushniak, S. (2022). Continued $\mathbf{A_2}$-fractions and singular functions. Matematychni Studii, 58(1), 3-12. https://doi.org/10.30970/ms.58.1.3-12
Section
Articles