On some questions related to the joint distribution of functionally dependent random variables

Consider the random vector $([sf_0(\eta)]_{m_0},\dots,[sf_N(\eta)]_{m_N})$ where $\eta$ is a random variable uniformly distributed on the interval $[0,2\pi]$; $s>0$ is a parameter and $[A]_m$ is the integral part of the least positive residue of a number $A$ modulo $m$. In the present paper, some classes of functions $f_0,\dots,f_N$ are found for which the distribution of this vector converges as $s\to\infty$ to the uniform distribution on integral points of $(N+1)$-dimensional rectangular
$$ \{x\in R^{N+1}\quad0\le x_i<m_i,\quad i=0,l,\dots,N\}. $$
Estimates of convergence rates are given.