We know that 2-interval exchange on $S_1$ is just a rotation on $S_1$, and there is a Rauzy–Veech–Zorich induction(Euclidean algorithm) or continue fraction to approximate the dynamic system very fast. in general, for $\alpha\in \mathbb{R}- \mathbb{Q}$ with bounded continue fraction coefficient, time average converage to space average in exponent speed, $$\frac{\{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset S_1\}}{n}=|b-a|+O(\log n)$$ and if $\alpha$ is type $\eta$, then, for every $\varepsilon>0,$ the discrepancy satisfies $$\frac{\{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset S_1\}}{n}=|b-a|+\mathrm{O}\left(n^{(-1 / \eta)+\varepsilon}\right) $$ The problem is, is it possible to gain a similar result for $n-$interval exchange map, more precisely, this split into two problem,

Problem1 Is it possible to define a type of $n$-interval exchange map?

Problem2 fix an $n$-interval exchange map $R_{(\lambda,\pi)}: \mathbb{T} \rightarrow \mathbb{T}, \pi\in \mathbb{S}_n$, if problem1 has an answer, is it possible to gain a discrepancy estimate for $R_{(\lambda,\pi)}$? i.e. $$\frac{\{k|1\leq k\leq n, \{R_{(\lambda,\pi)}^k(x)\}\in [a,b]\subset \mathbb{T}\}}{n}=|b-a|+\mathrm{O}\left(f(n)\right)$$ for genric $x\in \mathbb{T}$?

Definition. Let $\eta$ be a positive real number or infinity. The irrational number $\alpha$ is said to be of type $\eta$ if $\eta$ is the supremum of all $\gamma$ for which $\lim _{q \rightarrow \infty} q^{\gamma}\langle q \alpha\rangle=0,$ where $q$ runs through the positive integers. The problem is, is there a similar way to consider the type in $n$-interval exchange map for $b\geq 3$