Normal properties of numbers in the terms of their representation by the Perron series

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


UDC 511.7

We examine the representation of real numbers by the Perron series ($P$-representation) given by $$(0;1]\ni x=\sum\limits_{n=0}^{\infty}\frac{r_0 r_1\ldots r_n}{(p_1-1)p_1\ldots(p_n-1)p_n p_{n+1}}=\Delta^P_{p_1 p_2\ldots},\quad\mbox{where}\quad r_n,p_n\in\mathbb{N},\quad p_{n+1}\geq r_n+1,$$ and its transcoding ($\overline{P}$-representation) $$x=\Delta^{\overline{P}}_{g_1 g_2\ldots},\quad\mbox{where}\quad g_n=p_n-r_{n-1}.$$ We establish the properties of $\overline{P}$-representations typical of almost all numbers with respect to the Lebesgue measure (normal properties of the representations of numbers). We also examine the conditions of existence of the frequency of a digit $i$ in the $\overline{P}$-representation of a number $x=\Delta^{\overline{P}}_{g_1 g_2\ldots g_n\ldots}$ defined by the equality $$\nu_i^{\overline{P}}(x)=\lim\limits_{k\to\infty} \frac{N^{\overline{P}}_i(x,k)}{k},$$ where $N^{\overline{P}}_i(x,k)$ denotes the amount of numbers $n$ such that $g_n=i$ and $n\leq k.$ In particular, we establish conditions under which the frequency $\nu_i^{\overline{P}}(x)$ exists and is constant for almost all $x\in(0;1].$ In addition, we also determine the conditions under which the digits in $\overline{P}$-representations are encountered finitely or infinitely many times for almost all numbers from $(0;1].$


О. М. Барановський, М. В. Працьовитий, Б. І. Гетьман, Порівняльний аналіз метричних теорій представлень чисел рядами Енгеля і Остроградського та ланцюговими дробами, Наук. часопис Нац. пед. ун-ту ім. М. П. Драгоманова, Сер. 1, Фіз.-мат. науки, № 12, 130–139 (2011).

М. П. Мороз, Зображення дійсних чисел рядами Перрона, їхня геометрія та деякі застосування, Нелінійні коливання, 26, № 2, 247–260 (2023).

М. В. Працьовитий, Б. І. Гетьман, Ряди Енгеля та їх застосування, Наук. часопис Нац. пед. ун-ту ім. М. П. Драгоманова, Сер. 1, Фіз.-мат. науки, № 7, 105–116 (2006).

А. Я. Хинчин, Цепные дроби, Наука, Москва (1978).

O. Baranovskyi, M. Pratsiovytyi, One class of continuous functions with complicated local properties related to Engel series, Funct. Approx. Comment. Math. Adv. Publ., 1–20 (2022).

M. É. Borel, Les probabilités dénombrables et leurs applications arithmétiques, Rend. Circ. Mat. Palermo (1884-1940), 27, 247–271 (1909).

F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandl. d. 52 Versammlung deutscher Philologen und Schulmänner in Marburg, vol. 29, September bis 3. Oktober (1913), Leipzig, 190–191 (1914).

P. Erdős, A. Rényi, P. Szüsz, On Engel’s and Sylvester’s series, Ann. Univ. Sci. Budapest. Sect. Math., 1, 7–32 (1958).

J. Lüroth, Über eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Ann., 21, 411–423 (1883).

O. Perron, Irrationalzahlen, Walter de Gruyter & Co., Berlin (1960).

M. Pratsiovytyi, Yu. Khvorostina, Topological and metric properties of distributions of random variables represented by the alternating Lüroth series with independent elements, Random Oper. and Stoch. Equat., 21, № 4, 385–401 (2013).

M. V. Pratsiovytyi, Yu. V. Khvorostina, A random variable whose digits in the $widetilde{L}$-representation have the Markovian dependence, Theor. Probab. and Math. Statist., № 91, 157–168 (2015).

A. Rényi, A new approach to the theory of Engel’s series, Ann. Univ. Sci. Budapest. Sect. Math., 5, 25–32 (1962).

J. J. Sylvester, On a point in the theory of vulgar fractions, Amer. J. Math., 3, № 4, 332–335 (1880).

Yu. Zhykharyeva, M. Pratsiovytyi, Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers, Algebra and Discrete Math., 14, № 1, 145–160 (2012).

How to Cite
Moroz, M. “Normal Properties of Numbers in the Terms of Their Representation by the Perron Series ”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 7, July 2023, pp. 920 -32, doi:10.37863/umzh.v75i7.7503.
Research articles