Bounded remainder sets

We consider the category $(\mathcal{T,S,X})$ consisting of transformations $\mathcal{S\colon T\to T}$ of spaces $\mathcal T$ with distinguished subsets $\mathcal{X\subset T}$. Let $r_\mathcal X(i,x_0)$ be the distribution function of points from the $\mathcal S$-orbit $x_0,x_1=\mathcal S(x_0),\dots,x_{i-1}=\mathcal S^{i-1}(x_0)$ got in $\mathcal X$, and a deviation $\delta_\mathcal X(i,x_0)$ be defined by the equation
$$ r_\mathcal X(i,x_0)=a_\mathcal Xi+\delta_\mathcal X(i,x_0), $$
where $a_\mathcal Xi$ is the average value. If $\delta_\mathcal X(i,x_0)=O(1)$ then such $\mathcal X$ are called bounded remainder sets. In this article the bounded remainder sets $\mathcal X$ are built in the following cases: 1) the space $\mathcal T$ is a circle, a torus or a Klein bottle; 2) the map $\mathcal S$ is a rotation of the circle, a shift or an exchange transformation of the torus; 3) the $\mathcal X$ is a fixed subset $\mathcal{X\subset T}$ or a sequence of subsets depending on the iteration step $i=0,1,2,\dots$