Gauss–Kuzmin problem for the difference Engel-series representation of real numbers

  • M. P. Moroz Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev

Abstract

UDC 511.7+517.5

Let $x=\Delta^{\overline{E}}_{g_1(x)\ldots g_n(x)\ldots}$ be the difference Engel-series representation of a real number $x\in\left(0;1\right]$ (${\overline{E}}$-representation), where $\Delta^{\overline{E}}_{g_1\ldots g_n\ldots}=\displaystyle\sum\nolimits_{n=1}^\infty\dfrac{1}{(2+g_1)\ldots(2+g_1+\ldots+g_n)},$ $\omega^n(x)=\Delta^{\overline{E}}_{g_{n+1}(x)g_{n+2}(x)\ldots}$ is an $n$-fold  operator of left shift  of digits in the $\overline{E}$-representation of the number $x$.  For a sequence of sets $E_n(a)=\left\{x\colon x\in\left(0;1\right),\omega^n(x)<a\right\}$, where $a$ is a fixed parameter with $\left(0;1\right]$, it is proved that $\lim_{n\to\infty} \lambda\left(E_n(a)\right)=1$, where $\lambda(\cdot)$ is a Lebesgue measure. This problem is similar to the classical Gauss–Kuzmin problem for elementary continued  fractions. However, their solutions  are noticeably different.

References

B. I. Get'man, Zobrazhennya chisel s-adichnimi ryadami Engelya, Nauk. chasopis Nac. ped. un-tu im. M. P. Dragomanova. Ser. 1, Fiz.-mat. nauki, № 9, 212 – 224 (2008).

B. I. Get'man, Metrichni vlastivosti mnozhini chisel, viznachenih umovami na їh rozkladi v ryad Engelya, Nauk. chasopis Nac. ped. un-tu im. M. P. Dragomanova. Ser. 1, Fiz.-mat. nauki, № 10, 88 – 99 (2009).

R. O. Kuz'min, Ob odnoj zadache Gaussa, Dokl. AN SSSR, 375 – 380 (1928).

M. V. Prac'ovitij, B. I. Get'man, Ryadi Engelya ta їh zastosuvannya, Nauk. chasopis Nac. ped. un-tu im. M. P. Dragomanova. Ser. 1, Fiz.-mat. nauki, № 7, 105 – 116 (2006).

A. YA. Hinchin, Cepnye drobi, Nauka, Moskva (1978).

P. Levy, Sur les lois de probabilité dont dependent les quotients complets et incomplets d'une fraction continue. (French), Bull. Soc. Math. France, 57, 178 – 194 (1929). DOI: https://doi.org/10.24033/bsmf.1150

Published
09.08.2022
How to Cite
Moroz , M. P. “Gauss–Kuzmin Problem for the Difference Engel-Series Representation of Real Numbers”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 7, Aug. 2022, pp. 1004 -08, doi:10.37863/umzh.v74i7.7159.
Section
Short communications