I'm currently reading the chapter 4 of the book "Ergodic Theory with a view towards Number Theory" by Manfred Einsiedler and Thomas Ward.
More precisely I'm reading section 4.4 which deals with equidistribution. Then there is Weyl theorem:
Theorem: If $a_k$ is a irrational number the sequence $\{p(n)\}_{n\in \mathbb{N}}$, where $p(n)=a_k n^k+\cdots a_1 n+a_0$, is equidistributed in $S^1$, i.e, for any continuous function $f:S¹\to \mathbb{R}$ $$ \dfrac{1}{n}\sum_{n=0}^{\infty} f(p(n))\longrightarrow \int_{S^1} f(x)\; dLeb(x) $$ as $n\to \infty$.
The proof is based on the unique ergodicity of the skew product $$ T(x_1,\ldots, x_k)=(x_1+\alpha, x_2+x_1, \ldots, x_k+x_{k-1}) $$ on the torus $\mathbb{T}^k$. Then the autors shows that, $$ T(x_1,\ldots, x_k)= \left(\begin{array}{c} n\alpha+x_1 &\\ \binom{n}{2}\alpha+nx_1+x_2 &\\ \vdots &\\ \binom{n}{k}\alpha+ \binom{n}{k-1}x_1+\cdots+nx_{k-1}+x_k \end{array}\right) $$ So far no problem.
My problem: the autors claims that by putting $\alpha=k!a_k$ we can choose $x_1, \ldots, x_k$ such that $$ p(n)=\binom{n}{k}\alpha+ \binom{n}{k-1}x_1+\cdots+nx_{k-1}+x_k. $$ Therefore that is my difficult, how to choose those suitable $x_1, \ldots, x_k$ ?
Any polynomial $p(x)$ of degree at most $k$ can be written in the form $$p(x)=b_k\binom{x}{k}+b_{k-1}\binom{x}{k-1}+\dots+b_1\binom{x}{1}+b_0$$ where $\binom{x}{k}=\frac{x(x-1)\dots(x-k+1)}{k!}$ (so that for $n\in\mathbb N$ this definition coincides with the familiar one). This result can be established in a variety of ways, the most straightforward being by induction on $k$.
To see that $b_k$ is $k!$ times the leading coefficient of $p$, just note that $\binom{x}{k}$ has leading coefficient $\frac{1}{k!}$.