Канторівська двійково-фібоначчієва система числення у задачах теорії функцій

Abstract

In the article we study a class of continuous functions with locally complicated structure defined in terms of the representation of numbers in the Cantor number system:
\[[0;1]\ni x=\frac{\alpha_1}{s_1}+\frac{\alpha_2}{s_1s_2}+\cdots+\frac{\alpha_n}{s_1s_2 \ldots s_n}+\cdots \equiv\Delta_{\alpha_1\alpha_2 \ldots \alpha_n \ldots},\]
where $s_n=2^{\varphi_n}$, $(\varphi_n)$
is a classical Fibonacci sequence: $\varphi_1=1=\varphi_2$, \mbox{$\varphi_{n+2}=\varphi_n+\varphi_{n+1}$,} $\alpha_n\in \{0, 1, \ldots, s_n-1\}$.

Singular, nowhere monotonic and nondifferentiable functions, functions with bounded and unbounded variation are among the studied functions.