We know that 2-interval exchange on $\mathbb{T}$ is just a rotation on $\mathbb{T}$, and there is a Rauzy–Veech–Zorich induction (Euclidean algorithm or continued fraction) to approximate the dynamic system very fast. To mention Rauzy–Veech–Zorich induction is because the toy model of it, i.e. continued fraction can directly gain to following corollary (without estimating the exponential sum occuring in Weyl criterion)

Corollary of continued fraction in general, for $\alpha\in \mathbb{R}- \mathbb{Q}$ with bounded continued fraction coefficient, i.e. $\alpha=\left[a_{0}, a_{1}, a_{2}, \ldots\right], \sup_{i\in \mathbb{N}}a_i\leq +\infty$, time average converges 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 we have more than the corollary, that is,

Theorem 1 if $\alpha$ is type $\eta$, then, for every $\varepsilon>0,$ the discrepancy satisfies $$\frac{\{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset S_1\}}{n}=|b-a|+\mathrm{O}\left(n^{(-1 / \eta)+\varepsilon}\right) $$

Although I do not know whether Theorem 1 can directly gain from continued fraction, at least it could be gain from estimate the exponential sum occuring in Weyl criterion, so it is true at least. This lead me to consider if the situation is the similar for $n$-interval exchange. The problem, more precisely, split into two problem, one is should there be a "metric" could evaluate the behaviour of a generic $n$-interval exchange such as the below Definition1 for rotation($2$-interval exchange) do, the other one is, use this "type" (if it exist), could we get similar result as above for $n$-interval exchange.

Problem 1 Is it possible to define a type of $n$-interval exchange map?

Problem 2 fix an $n$-interval exchange map $R_{(\lambda,\pi)}: \mathbb{T} \rightarrow \mathbb{T}, \pi\in \mathbb{S}_n$, if problem 1 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 generic $x\in \mathbb{T}$?

Definition1. 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 $b\geq 3$, where, $$ \langle t\rangle=\min _{n \in \mathbb{Z}}|t-n|=\min (\{t\}, 1-\{t\}) $$