Van der curpurt trick

There is the statement of Van der carport theorem:

Given a sequences $ \{x_n\}_{n=1}^{\infty}$ in $ S_1$, if $ \forall k\in N^*$, $ \{x_{n+k}-x_n\}$ is uniformly distributed, then $ \{x_n\}_{n=1}^{\infty}$ is uniformly distributed.

I do not know how to establish this theorem with no extra condition, but this result is true at least for polynomial flow.

$ |\sum_{n=1}^Ne^{2\pi imQ(n)}|= \sqrt{(\sum_{n=1}^Ne^{2\pi imQ(n)})(\overline{\sum_{n=1}^Ne^{2\pi imQ(n)}})}$

$ = \sqrt{\sum_{h_1=1}^N\sum_{n=1}^{N-h_1}e^{2\pi imQ(n+h_1)-Q(n)}}=\sqrt{\sum_{h_1=1}^N\sum_{n=1}^{N-h_1}e^{2\pi im \partial^1_{h_1}Q(n)}} \leq \sqrt{\sum_{h_1=1}^N|\sum_{n=1}^{N-h_1}e^{2\pi \partial^1_{h_1}Q(n)}|}$

$ = \sqrt{\sum_{h_1=1}^N\sqrt{ (\sum_{n=1}^{N-h_1}e^{2\pi \partial^1_{h_1}Q(n)} )(\overline{\sum_{n=1}^{N-h}e^{2\pi \partial^1_{h_1}Q(n)})}}}\leq\sqrt{\sum_{h_1=1}^N\sqrt{ \sum_{h_2=1}^N|\sum_{n=1}^{N-h_1}e^{2\pi\partial^1_{h_2} \partial^1_hQ(n)} |}}$

$ \leq ….\leq$

$ \sqrt{\sum_{h_1=1}^N\sqrt{ \sum_{h_2=1}^N \sqrt{….\sqrt{\sum_{h_{k-1}=1}^{N-h_{k-2}}|\sum_{n=1}^{N-h_{k-1}}e^{2\pi\partial_{h_1h_2…h_{k-1}Q(n)}}|}}}} =o(1) $


This type of trick could also establish the following result, which could be understand as a discretization of the Vinegradov lemma.

Uniformly distribution result of $  F_p$:
Given $  Q(n)=a_kn^k+...+a_1n+a_0$, $  \{Q(0),Q(1),...,Q(p-1)\}$ coverages
 to a uniformly distribution in $  \{0,1,...,p-1\}$
 as $  p \to \infty$.

This trick could also help to establish estimate of correlation of low complexity sequences and multiplicative function, such as result:

$ S(x)=\sum_{n\le x}\left(\frac{n}{p}\right)\mu(n)=o(n)$

Maybe with the help of B-Z-S theorem.

The standard estimate of Mobius function is:

$ \sum_{n\leq X:n\equiv a~(mod~q)} \mu^2(n)=\frac{6}{\pi^2} \prod_{p|q} \left(1-\frac{1}{p^2} \right)\frac{X}{q}+E(X,q,a)$

The error term $ O_{\varepsilon}\left(\sqrt{X/q} +q^{\frac{1}{2}+\varepsilon}\right)$ is true for $ q\leq X^{\frac{2}{3}-\varepsilon}$.