I wish to understand a remark given in the book 'Higher order fourier analysis' by T. Tao. The remark is related to the following proposition:
Proposition 1.1.17 (Single-scale equidistribution theorem for abelian polynomial sequences). Let $P$ be a polynomial map from $\mathbf{Z}$ to $\mathbf{T}^{d}$ (torus of dimension $d$) of some degree $s \geq 0$, and let $F: \mathbf{R}^{+} \rightarrow \mathbf{R}^{+}$ be an increasing function. Then there exists an integer $1 \leq M \leq O_{F, s, d}(1)$ and a decomposition $$ P=P_{\text {smth }}+P_{\text {equi }}+P_{\text {rat }} $$ into polynomials of degree $s$, where
(i) $\left(P_{\mathrm{smth}}\right .$ is smooth) The $i^{\text {th }}$ coefficient $\alpha_{i, \mathrm{smth}}$ of $P_{\mathrm{smth }}$ has size $O\left(M / N^{i}\right)$. In particular, on the interval $[N]=\{1,2,\cdots, N\}, P_{\text {smth }}$ is Lipschitz with homogeneous norm $O_{s, d}(M / N)$.
(ii) ( $P_{\text {equi }}$ is equidistributed) There exists a subtorus $T$ of $\mathbf{T}^{d}$ of complexity at most $M$ and some dimension $d^{\prime}$, such that $P_{\text {equi }}$ takes values in $T$ and is totally $1 / F(M)$-equidistributed on $[N]$ in this torus (after identifying this torus with $\mathbf{T}^{d^{\prime}}$ using an invertible linear transformation of complexity at most $M)$.
(iii) $\left(P_{\text {rat }}\right.$ is rational) The coefficients $\alpha_{i,\text { rat }}$ of $P_{\text {rat }}$ are such that $q \alpha_{i,\text { rat }}$ $=0$ for some $1 \leq q \leq M$ and all $0 \leq i \leq s$. In particular, $q P_{\text {rat }}=0$ and $P_{\mathrm{rat}}$ is periodic with period $q$.
If, furthermore, $F$ is of polynomial growth, and more precisely $F(M) \leq$ $K M^{A}$ for some $A, K \geq 1$, then one can take $M \ll_{A, s, d} K^{O_{A, s, d}(1)}$.
Remark 1.1.20. We will see how this smooth-equidistributed-rational decomposition evolves as $N$ increases. Roughly speaking, the torus $T$ that the $P_{\text {equi }}$ component is equidistributed on is stable at most scales, but there will be a finite number of times in which a "growth spurt" occurs and $T$ jumps up in dimension.
For instance, consider the linear flow $P(n):=\left(n / N_{0}, n / N_{0}^{2}\right) \bmod \mathbf{Z}^{2}$ on the two-dimensional torus. At scales $N \ll N_{0}$ (and with $F$ fixed, and $N_{0}$ assumed to be sufficiently large depending on $F$ ), $P$ consists entirely of the smooth component. But as $N$ increases past $N_{0}$, the first component of $P$ no longer qualifies as smooth, and becomes equidistributed instead; thus in the range $N_{0} \ll N \ll N_{0}^{2}$, we have $P_{\mathrm{smth}}(n)=\left(0, n / N_{0}^{2}\right) \bmod \mathbf{Z}^{2}$ and $P_{\text {equi }}(n)=\left(n / N_{0}, 0\right) \bmod \mathbf{Z}^{2}$ (with $P_{\text {rat }}$ remaining trivial i.e. $P_{\text{rat}}=0$), with the torus $T$ increasing from the trivial torus $\{0\}^{2}$ to $\mathbf{T}^{1} \times\{0\}$.
A second transition occurs when $N$ exceeds $N_{0}^{2}$, at which point $P_{\text {equi }}$ encompasses all of $P$.
Evolving things in a somewhat different direction, if one then increases $F$ so that $F(1)$ is much larger than $N_{0}^{2}$, then $P$ will now entirely consist of a rational component $P_{\text {rat}}$.
Question 1: When $N \gg N_0$, how does it follow that the smooth component breaks down into smooth and equidistributed components?
Question 2: When $N$ exceeds $N_0^2$, why does the whole component becomes equidistributed?
Question 3: When we assume $F(1) \gg N_0^2$ , how come $P$ is entirely the rational component only?
Can someone please answer this question by making use of the lemma stated above and explain how things are moving relative to the size of $N_0^2, N$ and $F(1)$. Please help me in understanding this remark!
Thank you.