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We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) is just a rotation on $\mathbb{T}$, and there is a process called Euclidean algorithm or continued fraction to accelate the time to approximate the dynamic system's phase in very fast speed. For general $n$-interval exchange the process is called Rauzy–Veech–Zorich induction. As a toy model of it, continued fraction can directly gain the following corollary (without estimating the exponential sum occuring in Weyl criterion)

Corollary of continued fraction 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, $$\sup_{0<a<b<1}|\# \{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset \mathbb{T}\}-n(b-a)|=\mathrm{O}(\log n)$$

and we have more than the corollary, that is,

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

Although I do not know whether Theorem 1 can directly gain from continued fraction (if the Additional question has a positive answer, then Theorem 1 can be prove only with continued fraction), anyway at least it could be derived 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, that is, Rauzy–Veech–Zorich induction can help us gain a discrepancy estimate. 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.

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

question 2 fix an $n$-interval exchange map $R_{(\lambda,\pi)}: \mathbb{T} \rightarrow \mathbb{T}, \pi\in \mathbb{S}_n$, if question 1 has an answer, is it possible to gain a discrepancy estimate for $R_{(\lambda,\pi)}$? i.e. $$|\sup_{0<a<b<1}\# \{k|1\leq k\leq n, \{R_{(\lambda,\pi)}^k(x)\}\in [a,b]\subset \mathbb{T}\}-(b-a)|=\mathrm{O}\left(f(n)\right)$$ for generic $x\in \mathbb{T}$? and the $f(n)$ grow slower when the type of $R_{(\lambda,\pi)}$ is better.

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, $$ \langle t\rangle=\min _{n \in \mathbb{Z}}|t-n|=\min (\{t\}, 1-\{t\}) $$

Additional question is it true that irrational number $\alpha=\left[a_{0}, a_{1}, a_{2}, \ldots\right]$ has type $\eta$ iff $$\sum_{i=0}^{\infty}\frac{a_i}{n^{\eta+\epsilon}}$$ converage for all $\epsilon>0$?

We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) is just a rotation on $\mathbb{T}$, and there is a process called Euclidean algorithm or continued fraction to accelate the time to approximate the dynamic system's phase in very fast speed. For general $n$-interval exchange the process is called Rauzy–Veech–Zorich induction. As a toy model of it, continued fraction can directly gain the following corollary (without estimating the exponential sum occuring in Weyl criterion)

Corollary of continued fraction 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, $$\sup_{0<a<b<1}|\# \{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset \mathbb{T}\}-n(b-a)|=\mathrm{O}(\log n)$$

and we have more than the corollary, that is,

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

Although I do not know whether Theorem 1 can directly gain from continued fraction (if the Additional question has a positive answer, then Theorem 1 can be prove only use continued fraction), anyway at least it could be derived 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, that is, Rauzy–Veech–Zorich induction can help us gain a discrepancy estimate. 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.

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

question 2 fix an $n$-interval exchange map $R_{(\lambda,\pi)}: \mathbb{T} \rightarrow \mathbb{T}, \pi\in \mathbb{S}_n$, if question 1 has an answer, is it possible to gain a discrepancy estimate for $R_{(\lambda,\pi)}$? i.e. $$|\sup_{0<a<b<1}\# \{k|1\leq k\leq n, \{R_{(\lambda,\pi)}^k(x)\}\in [a,b]\subset \mathbb{T}\}-(b-a)|=\mathrm{O}\left(f(n)\right)$$ for generic $x\in \mathbb{T}$? and the $f(n)$ grow slower when the type of $R_{(\lambda,\pi)}$ is better.

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, $$ \langle t\rangle=\min _{n \in \mathbb{Z}}|t-n|=\min (\{t\}, 1-\{t\}) $$

Additional question is it true that irrational number $\alpha=\left[a_{0}, a_{1}, a_{2}, \ldots\right]$ has type $\eta$ iff $\eta$ is the supremum such that, $$\sum_{i=0}^{\infty}\frac{a_i}{n^{\eta-\epsilon}}$$ diverage for all $\epsilon>0$?

We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) is just a rotation on $\mathbb{T}$, and there is a process called Euclidean algorithm or continued fraction to accelate the time to approximate the dynamic system's phase in very fast speed. For general $n$-interval exchange the process is called Rauzy–Veech–Zorich induction. As a toy model of it, continued fraction can directly gain the following corollary (without estimating the exponential sum occuring in Weyl criterion)

Corollary of continued fraction 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, $$\sup_{0<a<b<1}|\# \{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset \mathbb{T}\}-n(b-a)|=\mathrm{O}(\log n)$$

and we have more than the corollary, that is,

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

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, that is, Rauzy–Veech–Zorich induction can help us gain a discrepancy estimate. 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. $$|\sup_{0<a<b<1}\# \{k|1\leq k\leq n, \{R_{(\lambda,\pi)}^k(x)\}\in [a,b]\subset \mathbb{T}\}-(b-a)|=\mathrm{O}\left(f(n)\right)$$ for generic $x\in \mathbb{T}$? and the $f(n)$ grow slower when the type of $R_{(\lambda,\pi)}$ is better.

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, $$ \langle t\rangle=\min _{n \in \mathbb{Z}}|t-n|=\min (\{t\}, 1-\{t\}) $$

We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) is just a rotation on $\mathbb{T}$, and there is a process called Euclidean algorithm or continued fraction to accelate the time to approximate the dynamic system's phase in very fast speed. For general $n$-interval exchange the process is called Rauzy–Veech–Zorich induction. As a toy model of it, continued fraction can directly gain the following corollary (without estimating the exponential sum occuring in Weyl criterion)

Corollary of continued fraction 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, $$\sup_{0<a<b<1}|\# \{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset \mathbb{T}\}-n(b-a)|=\mathrm{O}(\log n)$$

and we have more than the corollary, that is,

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

Although I do not know whether Theorem 1 can directly gain from continued fraction (if the Additional question has a positive answer, then Theorem 1 can be prove only with continued fraction), anyway at least it could be derived 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, that is, Rauzy–Veech–Zorich induction can help us gain a discrepancy estimate. 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.

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

question 2 fix an $n$-interval exchange map $R_{(\lambda,\pi)}: \mathbb{T} \rightarrow \mathbb{T}, \pi\in \mathbb{S}_n$, if question 1 has an answer, is it possible to gain a discrepancy estimate for $R_{(\lambda,\pi)}$? i.e. $$|\sup_{0<a<b<1}\# \{k|1\leq k\leq n, \{R_{(\lambda,\pi)}^k(x)\}\in [a,b]\subset \mathbb{T}\}-(b-a)|=\mathrm{O}\left(f(n)\right)$$ for generic $x\in \mathbb{T}$? and the $f(n)$ grow slower when the type of $R_{(\lambda,\pi)}$ is better.

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, $$ \langle t\rangle=\min _{n \in \mathbb{Z}}|t-n|=\min (\{t\}, 1-\{t\}) $$

Additional question is it true that irrational number $\alpha=\left[a_{0}, a_{1}, a_{2}, \ldots\right]$ has type $\eta$ iff $$\sum_{i=0}^{\infty}\frac{a_i}{n^{\eta+\epsilon}}$$ converage for all $\epsilon>0$?

We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) 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, $$\sup_{0<a<b<1}|\# \{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset \mathbb{T}\}-n(b-a)|=O(\log n)$$

and we have more than the corollary, that is,

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

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. $$|\sup_{0<a<b<1}\# \{k|1\leq k\leq n, \{R_{(\lambda,\pi)}^k(x)\}\in [a,b]\subset \mathbb{T}\}-(b-a)|=\mathrm{O}\left(f(n)\right)$$ for generic $x\in \mathbb{T}$? and the $f(n)$ grow slower when the type of $R_{(\lambda,\pi)}$ is better.

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, $$ \langle t\rangle=\min _{n \in \mathbb{Z}}|t-n|=\min (\{t\}, 1-\{t\}) $$

We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) is just a rotation on $\mathbb{T}$, and there is a process called Euclidean algorithm or continued fraction to accelate the time to approximate the dynamic system's phase in very fast speed. For general $n$-interval exchange the process is called Rauzy–Veech–Zorich induction. As a toy model of it, continued fraction can directly gain the following corollary (without estimating the exponential sum occuring in Weyl criterion)

Corollary of continued fraction 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, $$\sup_{0<a<b<1}|\# \{k|1\leq k\leq n, \{k\alpha\}\in [a,b]\subset \mathbb{T}\}-n(b-a)|=\mathrm{O}(\log n)$$

and we have more than the corollary, that is,

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

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, that is, Rauzy–Veech–Zorich induction can help us gain a discrepancy estimate. 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. $$|\sup_{0<a<b<1}\# \{k|1\leq k\leq n, \{R_{(\lambda,\pi)}^k(x)\}\in [a,b]\subset \mathbb{T}\}-(b-a)|=\mathrm{O}\left(f(n)\right)$$ for generic $x\in \mathbb{T}$? and the $f(n)$ grow slower when the type of $R_{(\lambda,\pi)}$ is better.

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, $$ \langle t\rangle=\min _{n \in \mathbb{Z}}|t-n|=\min (\{t\}, 1-\{t\}) $$