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分类： Vinogradov estimate

  • Vinogradov 估计的一条思路：环面均匀分布与连分数尺度

    旧博客原文

    原题：An approach to Vinogradov estimate

    Vinogradov estimate is:

    |\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

    补充说明

    以下是新整理的中文说明；上方旧博客原文保持不变。

    Vinogradov 型估计可以看成带有有效误差的均匀分布定理。若相位含有无理参数，问题往往在两个语言之间切换：动力系统上是环面轨道的均匀分布，解析数论上是指数和的消失。

    Vinogradov 估计的一条思路：环面均匀分布与连分数尺度
    Vinogradov 型估计可以看成环面轨道均匀分布的定量版本，核心是控制指数和。

    1. 动力系统图像

    考虑环面上的序列

    $$n\mapsto (n\alpha,n^2\alpha,\ldots)\pmod1.$$

    当 $\alpha$ 无理时，这类序列常常均匀分布。Vinogradov 估计要求更强：不仅要知道平均极限，还要给出可用的误差项。

    2. 指数和形式

    Weyl criterion 把均匀分布转成指数和：

    $$\frac1N\sum_{n\le N}e(P(n))\to0.$$

    有效估计则要控制

    $$\left|\sum_{n\le N}e(P(n))\right|.$$

    当 $P$ 的系数包含无理数时，连分数近似会决定哪些尺度上相位最接近有理、哪些尺度上可以得到抵消。

    3. Fiber 均匀分布的限制

    一种诱人的想法是先证明每个 fiber 上的均匀分布，再合成整个空间的均匀分布。但这并不总是成立：所有纤维方向看起来平均，不代表整体分布没有隐藏相关性。这个失败提示我们必须直接控制整体指数和。

    4. 连分数尺度

    设 $\alpha$ 的 convergents 是 $p_k/q_k$。在长度接近 $q_k$ 的区间上，$n\alpha$ 的分布有特别好的结构。把 $[1,N]$ 拆成这些标准尺度，可以把任意长度的问题化成一族可估的块。

    5. 证明路线

    一个合理策略是：先用 Weyl differencing 降低多项式相位阶数；再用连分数控制主要尺度；最后把误差在不同块上求和。动力系统语言提供几何直觉，真正的定量估计则来自指数和技术。