\(\alpha\)-additive functions and uniform distribution modulo one (Q761495)

Every nonnegative integer n can be written in the form \(n=\sum^{\infty}_{k=0}\epsilon_ k(n)z_ k\), where \(z_ k\) is the sequence of denominators of the convergents of the continued fraction expansion of a given irrational \(\alpha =[a_ 0;a_ 1,a_ 2,...]\), \(\epsilon_ 0(n)\in \{0,1,...,a_ 1-1\}\), \(\epsilon_ k(n)\in \{0,1,...,a_{k+1}\}\) (k\(\geq 1)\), \(\epsilon_{k-1}(n)=0\) if \(\epsilon_ k(n)=a_{k+1}\). \textit{J. Coquet} [Bull. Soc. R. Sci. Liège 51, 161-165 (1982; Zbl 0497.10040)] proved for irrationals x that the sequence (x f(n))\({}^{\infty}_{n=0}\) is u.d. (uniformly distributed) mod 1, if f(n) denotes an \(\alpha\)-additive function, i.e. \(f(n)=\sum^{\infty}_{k=0}f(\epsilon_ k(n)z_ k).\) In the present article the following generalization is given: Let \(\phi\) : \({\mathbb{N}}_ 0\to {\mathbb{R}}\) be a function such that \(\phi (0)=0\), \(\phi\) (1) is irrational and \(f(n)=\sum^{\infty}_{k=0}\phi (\epsilon_ k(n))\). Then (f(n)) is u.d. mod 1 if (\(\phi\) (n)) is u.d. mod 1.