Uniform distribution and fractal dimensions for generalized Cantor products
For any \(x\in[0,1)\) and \(k\geq 1\), let \(x=\prod_{i=0}^\infty\frac{r_i(x)}{r_i(x)+k}\) be a generalized Cantor product. The Hausdorff dimension of certain sets concerning \((r_n(x))_{n\geq 0}\) are considered. Let \((t_n)_{n\geq 0}\) be defined by \[ t_n(x)=\frac{T^{(n+1)}(x)-b_{r_n(x)}}{1-b_{r_n(x)}} =\frac{T^{(n)}(x)-a_{r_n(x)}}{a_{r_n(x)+1}-a_{r_n(x)}},\quad x\in[0,1) n\geq 0, \] where \(T(x)=\frac{r_0(x)+k}{r_0(x)}x\) with \(r_0(x)=[\frac{kx}{1-x}]+1\), \(a_m=\frac{m-1}{m+k-1}\), \(b_m=a_m/a_{m+1}\) (\(m\geq 1\)) and \(T^{(n)}\) denotes the \(n\)th iterate of \(T\). \textit{Y. Lacroix} [Acta Arith. 63, 61-77 (1993; Zbl 0774.11042)] showed that for any \(x\in[0,1)\) this adjusted sequence \((t_n)_{n\geq 0}\) is completely uniformly distributed modulo \(1\). It is proved in this paper that values of \(x\in[0,1)\) such that \((t_n)_{n\geq 0}\) is not uniformly distributed modulo \(1\) is a set of Hausdorff dimension \(1\).
scientific article; zbMATH DE number 1747324
scientific article