每日归档： 2017年12月1日

Van der curpurt trick

hxypqr留下评论

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}.

 

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