$|\sum_{n=1}^{N}e^{2\pi i\alpha P(n)}|\leq c_A\frac{N}{log^A N}$

For fix $\alpha$ is irrational and $\forall A>0 … (*)$.

Assume $deg(P)=n$, this could view as a effective uniformly distribute result of dynamic system:  $([0,1]^n,T)$, where $T: x\to (A+B)x$, $b$ is a nilpotent matrix, matrix $A$ is identity but with a irrational number $\alpha$ in the $(n, n)$ elements.

First approach

we could easily to get a “uniform distribute on fiber” result without very much tough estimate to attach the theorem. That is just a application by my  “rigid trick” that is describe in my early note. But this approach is according to the understanding of the result as a uniformly distribute result on Torus $T^n$, we could do this approach with the last $S^1$, which will corresponding to $\partial^{n-1}x_k$, i.e. we could apply the “rigid trick” to prove sequences $(x_k,\partial^1 x_k,…, \partial^{n-1} x_k)$ is uniformly distribute according to $\partial^{n-1} x_k\in S^1$ .

Graph

But this approach seems difficult to generate. The difficulty is come from both there is no  similar uniformly distribute of the other perimeter use the rigid trick (At least as I know, I try to prove there could be one but I failed) and if in the best case we have the similar uniformly distribute result for other perimeter there is still some thing more need to be established. See this graph for a counterexample that the uniformly distribute for all fiberation could not derive a uniformly distribute for the original space.

Second approach

In this approach we need use the information of continue fractional to get some information (Which is of course critical to get some information about the estimate). But I do not know if it is necessary, maybe this could be a interesting question weather the information come from continue fractional must involve to get such a estimate in the future, but not today.

Any way, there is two different type of continue fractional:

1.$\alpha=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+…}}}$.

2.$\alpha=q_0+\frac{1}{q_1}+\frac{1}{q_1q_2}+\frac{1}{q_1q_2q_3}+\frac{1}{q_1q_2q_3q_4}+…$.

Anyway, these could be understand as a same thing more or less (if fact we can calculate some quantitive with $a_i,q_i$ which is roughly the same). That is just the orbits $\{e^{2\pi i\alpha}\}$ have quasi-period property, that is to say, under certain norms, it could be understand as the limits of periodic sequences. So it is natural to approximation $\{e^{2\pi i\alpha}\}$ by periodic sequences and will lead to a very good point-wise coverage result:

$T_k^{n}(x) \longrightarrow T^{n}(x)$

Where $T_k^{n}(x)=e^{2\pi i\sum_{i=1}^k\frac{1}{p_1…p_i}}$ is just the periodic approximation sequence which come from the best approximation (critical point of $||\frac{q}{p}-\alpha||$), which natural occur in continue fractional. And by this we already arrive a non qualitative form result of $(*)$ with $deg(P)=1$.

But unfortunately this approximation is too good to be true for $deg(P)\geq 2$ case. The reason of this result could be true is just because the natural estimate for the best approximation of $\alpha$; i.e. Dirichlet approximation theorem.

But for higher degree case, although we could not expect this thing to be true, we still could image a weaker but enough result to be true:

$\{T_k^{n}(x)\} \longrightarrow \{T^{n}(x)\}$

in the Gromov Hausdorff metric sense, and the $T_k^n(x)$ is carefully chose, which have a finite torsion structure(which could be view as a multilinear structure which will play a central role in the estimate). Here is a graph for $deg(P)=2$:

Roughly speaking,  in general $deg(P)=n$ case, there is a cube structure in the orbits $e^{2\pi iP(n\alpha)}$ and is critical to observe that the progression of difference structure in it. The goal of this approach is to establish some result from the finite torsion structure(multilinear structure). That is to say, the boundary is high order thing in all direction but there is only one direction attend to infinity the other is just a finite torsion, and we wish to get more information from the extra structure.

This also have a physics explaining, for which see the graph:

Third approach

For $P(n)=an^2+bn+c$ case:

$P(k+\Delta)=P(k)+(ka+b)\Delta+\Delta^2$