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一个 determinantal formula：Gram identity、Andreief identity 与对称多项式

旧博客原文

原题：A determinantal formula

I see a similar formula I wish to be true and merely have a proof in mind occur as a MO’s problem:

In my research, I encounter the following formula which I believe is correct (checked for $n\le3$ ). Is it classical ?

I am given a real symmetric matrix
$S:=\int Y(t)Y(t)^Td\mu(t),$
where $\mu$ is a probability and $Y(t):\Omega\rightarrow{\mathbb R}^n$ .

Let $\sigma_k(S)$ be the elementary symmetric polynomial in the eigenvalues of $S$ . For instance, $\sigma_1(S)$ is the trace and $\sigma_n(S)$ the determinant. The following formula gives $\sigma_k(S)$ in terms of the Gram matrix $G_k(s_1,\ldots,s_k)$ whose entries are the scalar products $Y(s_i)\cdot Y(s_j)$ .

$\sigma_k(S)=\frac1{k!}\int^{\otimes k}\det G_k(s_1,\ldots,s_k)\,d\mu(s_1)\cdots d\mu(s_k).$

Remark that $S$ is positive semi-definite. The integrand is non-negative, as well as $\sigma_k(S)$ . The integrand vanishes identically iff $Y(t)$ takes values in a subspace of dimension $<k$ , which is the condition under which $\sigma_k(S)$ vanishes. It follows that, if the formula above failed, it would be because of an inequality between strictly positive numbers.

The case $k=n$ is a consequence of the identity

$\int \det(f_j(s_k))\det(g_j(s_k))\prod_{j=1}^N d\mu(s_j) = N!\ \det\left(\int d\mu(t) f_j(t)g_k(t)\right)$

which I have seen under the names “Andreief identity” and also “Gram identity”. The proof is elementary using the Leibniz formula for the determinant.

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。

很多行列式公式可以理解为 Gram identity 或 Andreief identity 的变体。它们把矩阵的对称多项式、向量组的体积和积分中的 determinant 联系起来。

Gram identity 和 Andreief identity 把行列式、体积平方和对称多项式联系起来。

1. Gram matrix

给定向量 $v_1,\ldots,v_m$，Gram matrix 为

$$G_{ij}=\langle v_i,v_j\rangle.$$

$\det G$ 等于这些向量张成平行体体积的平方。因此 $G$ 半正定，且 $\det G=0$ 当且仅当向量线性相关。

2. Andreief identity

Andreief identity 的典型形式是

$$\int \det(f_i(x_j))\det(g_i(x_j))\prod_j d\mu(x_j)
=n!\det\left(\int f_i(x)g_j(x)d\mu(x)\right).$$

它是积分版的 Cauchy-Binet 公式。

3. 对称多项式

若 $A$ 是半正定矩阵，特征值为 $\lambda_1,\ldots,\lambda_n$，第 $k$ 个 elementary symmetric polynomial 是

$$e_k(A)=\sum_{i_1<\cdots
它也等于所有 $k\times k$ 主子式之和。

4. 几何解释

$e_k(A)$ 可以看成 $A$ 在所有 $k$ 维方向上的体积膨胀总和。若 $A$ 来自随机向量的 Gram matrix，那么对应公式会把 $e_k(A)$ 写成某种 determinant 的积分平均。

5. 证明方式

这类公式通常从 Leibniz determinant expansion 出发，交换求和和积分，再识别为 Cauchy-Binet 或 Andreief identity。核心不是计算技巧，而是“行列式等于体积平方”的几何含义。

2017年11月17日
等周不等式：Steiner 对称化、Minkowski-Steiner 公式与变分图像

旧博客原文

原题：Isoperimetric inequality

Introduction

the statement go isometry inequality is very simple:

$\Omega\subset R^n$ , iff $\Omega$ is a ball, $\frac{Vol(\Omega)}{Surf(\Omega)}$ arrive a minimum .

This is a classical problem in variation theory. The difficult is divide into two parts. The first is to create a “flow” which descrement the energy and the “flow” is compatible with the feature of a ball, i.e. every set under the flow will tend to like a “ball”. The second one is to proof there exist a unit in the space $surf(\Omega)=constant>0$ make the Energy $E(\Omega)=Vol(\Omega)$ arrive a minimum.

Combine this two property we can consult that ball is the set and definitely the only set make the $\frac{Vol(\Omega)}{Surf(\Omega)}$ arrive the minimum.

first difficulties

The energy $E(\Omega)$ is scaling invariance. The first difficult could divide into two part:

Restrict to convex set

the first is to deform a set into a convex set and proof this process would not lower $\frac{Vol(\Omega)}{Surf(\Omega)}$ . This could been a little subtle. and the way I image could make sense is just like the following transform:

but this process is harder in higher dimension, for example:

Convex set to a ball

Steiner symmetric process.

Affine transform

Minkowski–Steiner formula
In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the “derivative” of enclosed volume in an appropriate sense.

The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner.

Statement of the Minkowski-Steiner formula

Let $n \geq 2$ , and let $A \subsetneq \mathbb{R}^{n}$ be a compact set. Let $\mu (A)$ denote the [[Lebesgue measure]] (volume) of $A$ . Define the quantity $\lambda (\partial A)$ by the ”’Minkowski–Steiner formula”’:

$\lambda (\partial A) := \liminf_{\delta \to 0} \frac{\mu \left( A + \overline{B_{\delta}} \right) - \mu (A)}{\delta}$

where:

$\overline{B_{\delta}} := \left\{ x = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n} \left| | x | := \sqrt{x_{1}^{2} + \dots + x_{n}^{2}} \leq \delta \right. \right\}$

denotes the [[closed ball]] of [[radius]] $\delta > 0$ , and:

$A + \overline{B_{\delta}} := \left\{ a + b \in \mathbb{R}^{n} \left| a \in A, b \in \overline{B_{\delta}} \right. \right\}$

is the [[Minkowski sum]] of $latexA$ and $\overline{B_{\delta}}$ , so that:

$A + \overline{B_{\delta}} = \left\{ x \in \mathbb{R}^{n} | |x - a| \leq \delta \mbox{ for some } a \in A \right\}$ .

Surface measure

For “sufficiently regular” sets $A$ , the quantity $\lambda (\partial A)$ does indeed correspond with the $(n - 1)$ -dimensional measure of the [[boundary (topology)|boundary]] $\partial A$ of $A$ . See Federer (1969) for a full treatment of this problem.

Convex sets

When the set $A$ is a [[convex set]], the [[limit inferior|lim-inf]] above is a true [[Limit of a sequence|limit]], and one can show that

: $\mu \left( A + \overline{B_{\delta}} \right) = \mu (A) + \lambda (\partial A) \delta + \sum_{i = 2}^{n - 1} \lambda_{i} (A) \delta^{i} + \omega_{n} \delta^{n}$ ,

where the $\lambda_{i}$ are some [[continuous function]]s of $A<$ (see [[quermassintegral]]s) and $\omega_{n}$ denotes the measure (volume) of the [[unit ball]] in $\mathbb{R}^{n}$ :

: $\omega_{n} = \frac{2 \pi^{n / 2}}{n \Gamma (n / 2)}$ ,

where $\Gamma$ denotes the [[Gamma function]].

==Example: volume and surface area of a ball==

Taking $A = \overline{B_{R}}$ gives the following well-known formula for the surface area of the [[sphere]] of radius $R$ , $S_{R} := \partial B_{R}$ :

: $\lambda (S_{R}) = \lim_{\delta \to 0} \frac{\mu \left( \overline{B_{R}} + \overline{B_{\delta}} \right) - \mu \left( \overline{B_{R}} \right)}{\delta}$
:: $= \lim_{\delta \to 0} \frac{[ (R + \delta)^{n} - R^{n} ] \omega_{n}}{\delta}$
:: $= n R^{n - 1} \omega_{n}$ ,

where $\omega_{n}$ is as above.

something with more details

The second difficulties

To establish a continue property of the Energy functional $E(\Omega)= \frac{Vol(\Omega)}{Surf(\Omega)}$ .

the continuous property is consider with all open set $\Omega$ with Gromov-Hausdorff metric $d(\Omega_1,\Omega_2)= \inf_{metric\ d on \Omega_1 \cup \Omega_2}\sup_{x_1\in \Omega_1, x_2\in \Omega_2}d(x_1,x_2)$ .

We need to proof the continuous of $E(\Omega)$ with the Gromov-hausdorff metric on the space consist with convex open sets.

To remark,we need to observe that polygon approximation is just corresponding to the $\delta-seperate$ points approximation in Gromov-hausdorff distance. and definitely carefully refinement of this kind of approximation could lead to the result of continuous of the energy $E(\Omega)$ on convex set.

A second remark, we definitely need a definition of the $surf(\Omega)$ it could be achieve with open convex set $\Omega$ by a outer and inter approximation by polygon and the error term estimate.

Further remark, isoperimetric inequality is a general phenomenon.

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。

等周不等式说，在给定体积的集合中，球的边界面积最小。它是变分法、凸几何和几何测度论共同的基本定理。

等周不等式说明固定体积下球最小化 perimeter，可由对称化或 Brunn-Minkowski 思想证明。

1. 基本陈述

在 $\mathbb R^n$ 中，等周不等式为

$$P(\Omega)^n\ge n^n\omega_n |\Omega|^{n-1},$$

等号当且仅当 $\Omega$ 是球。这里 $P(\Omega)$ 是 perimeter，$\omega_n$ 是单位球体积。

2. 变分图像

一个自然想法是构造能量下降 flow，使任意集合逐渐变圆，并且不增加 perimeter、不改变体积。真正困难在于：一般集合可能很粗糙，flow 的存在性和紧性都需要处理。

3. Steiner 对称化

Steiner symmetrization 沿一个方向把每条平行线上的截面替换成居中的区间。它保持体积，并且不增加 perimeter。重复对称化后，集合越来越接近球。

这提供了一个几何证明路线：先把集合变得越来越对称，再用紧性取极限。

4. Minkowski-Steiner 公式

对足够好的集合，考虑外平行体

$$\Omega_t=\Omega+tB.$$

Minkowski-Steiner 公式把 $|\Omega_t|$ 展开成 $t$ 的多项式，其一次项与 perimeter 相关：

$$\left.\frac{d}{dt}\right|_{t=0}|\Omega+tB|=P(\Omega).$$

配合 Brunn-Minkowski 不等式，可以推出等周不等式。

5. 核心困难

等周问题的困难可以分成两部分：一是证明极小值存在，二是证明极小者只能是球。前者需要紧性和 lower semicontinuity，后者需要对称化、Euler-Lagrange 方程或 Brunn-Minkowski 凹性。

2017年11月16日
k-Hessian 方程与 k-curvature 方程：椭圆性、测度与 Wolff 势

旧博客原文

原题：k-hessian equation and k-curvature equation

here is the problem, how to understand k-hessian equation and k-curvature equation.

k-hessian equation

k-hessian equation is:

$H_k(u)=\sigma_k(D^2(u))=f$ (*)

where u is admissible, i.e. $\forall 1\leq i\leq k$ , $\sigma_i(D^2(u))\geq 0$ . this is just the condition to make (*) be a elliptic equation.

The most important result is the following three:

1.sovable (*) with direchlet boundary condition.

This is mainly the contribution of Caffaralli in 90’s. According flexible function and maximum principle we can establish the $C^{1,\alpha}$ estimate and $C^{2,\alpha}$ estimate in the inter. And the $C^{2,\alpha}$ estimate near the boundary is establish according to the conformation invariant and some perbutation of the solution of k-hessian equation after special rescaling.

2.Hessian measure.

This is mainly the work of X.J.Wang and Trudinger. they proved:

in the meaning of viscosity solution, if $\sigma_k(D^2(u))=f$ . then we can associate a measure $\mu$ with $u$ ,and the following is right:

when $u\in C^2(\Omega)$ , $\mu(B_r(x))=\int_{B_r{x}}\sigma_k(D^2(u))$ .

if $u_1,..,u_n,...$ coverage to $u$ . then $\mu_1,...,\mu_n,...$ coverage to $\mu$ in weak sense.

this is merely depend on a priori estimate on $u$

3.pointwise estimate corresponding wolff potential.

the Wolff potential is:

$W^{\mu}_{k}(x,r)= \int_{0}^r(\frac{\mu(B_t(x))}{t^{n-2k}})^{\frac{1}{k}}\frac{1}{t}dt$

We can easily use rescaling to understand the reasonable of this potential, and use this potential Lubutin establish the following pointwise estimate:

$u\in \Phi_k(B_{4R}(x))$ , $u\leq 0$ , then we have:

$W^{\mu}_k(x,\frac{R}{2})\leq |u(0)| \leq W^{\mu}_k(x,2R)-sup_{B_{2R}}|u|$

the RHS could look as a corollary of classical A-B-P estimate. the LHS need combine several observation. mean-value property and some else.

This result could use to establish some result on singularity point can be removable.

k-curvature equation

1.sovable (*) with direchlet boundary condition.

This is also established by cafferalli.

2.curvature measure.

This is established very recently. mean curvature equation in 2014, by perron lift and modified, general case in 2016 by more complex calculate and method.

3.pointwise estimate corresponding wolff potential.

This still do not established, and is the main thing I focus on. Due to we can look as k-curvature as a “projection” of k-hessian equation, Calderon-Zegmund decomposition and the estimate of k-hessian equation maybe useful.

My ideas

look is as “average” of “loop space”, “surface space”.

1.Grassmannian bundle

n algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:
$p:G_{d}(E)\to X$
such that the fiber

$p^{-1}(x)=G_{d}(E_{x})$ is the Grassmannian of the d-dimensional vector subspaces of $E_x$ . For example,

$G_{1}(E)=\mathbb {P} (E)$ is the projective bundle of E. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme.

Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle S and universal quotient bundle Q that fit into

$0\to S\to p^{*}E\to Q\to 0$ .
Specifically, if V is in the fiber p−1(x), then the fiber of S over V is V itself; thus, S has rank $r = rk(E)$ and

$\wedge ^{r}S$ is the determinant line bundle. Now, by the universal property of a projective bundle, the injection

$\wedge ^{r}S\to p^{*}(\wedge ^{r}E)$ corresponds to the morphism over X:
$G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)$ ,
which is nothing but a family of Plücker embeddings.

The relative tangent bundle $T Gd(E)/X$ of $Gd(E)$ is given by[1]
$T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,$
which is morally given by the second fundamental form. In particular, when d = 1, the early exact sequence tensored with the dual of S = O(-1) gives:
$0\to {\mathcal {O}}_{\mathbb {P} (E)}\to p^{*}E\otimes {\mathcal {O}}_{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0$ ,
which is the relative version of the Euler sequence.

2.Explain of the fully nonlinear elliptic equation

Now, we could consider the determination $\sum_{i_1,...,i_k\in\{1,...,n\}}det(u_{ij})_{i,j\in \{i_1,...,i_k\}\times\{i_1,...,i_k\}}$ as the determination of transform: $(u_{i_1},...,u_{i_k}) \longrightarrow (e_{i_1},...,e_{i_k})$ .

Now we need to understand $\sigma_k(D^2(u))=f$ at a point $x_0$ as the average of determination of transform matrix of $(u_{i_1},...,u_{i_k}) \longrightarrow (e_{i_1},...,e_{i_k})$ on Grassmannian manifold $G_k(x_0)$ is equal to $f(x_0)$ , i.e.:

$\int_{G_k(x_0)} det(\frac{\partial u_{i_a}}{\partial e_{i_b}}) d\mu=f(x_0)$

where $\mu$ is the natural haar measure on $G_k(x_0) \simeq G_k$ .

But the difficult to make the argument rigorous is that $u_i$ is scale and $e_i$ is vector.

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。

$k$-Hessian 方程是完全非线性椭圆方程中的基本模型。它把 Hessian 矩阵特征值的第 $k$ 个基本对称函数作为主算子；椭圆性不再自动成立，而是依赖 admissible cone。

$k$-Hessian 方程的椭圆性来自 admissible cone，弱解理论则进入 Hessian measure 和势估计。

1. 方程与 admissibility

设 $\lambda(D^2u)$ 是 Hessian 的特征值，定义

$$S_k(D^2u)=\sigma_k(\lambda(D^2u)).$$

$k$-Hessian 方程写作

$$S_k(D^2u)=f.$$

为了让方程椭圆，需要要求

$$\lambda(D^2u)\in \Gamma_k=\{\sigma_1>0,\ldots,\sigma_k>0\}.$$

这样的 $u$ 称为 $k$-admissible。这个条件是非线性椭圆理论的入口。

2. Dirichlet 问题

Dirichlet 问题要求

$$S_k(D^2u)=f\quad\text{in }\Omega,\qquad u|_{\partial\Omega}=\varphi.$$

经典策略是建立 $C^0$、梯度和二阶先验估计，再用连续性方法。边界估计通常最细，需要利用 domain 的几何条件、barrier function 和 rescaling。

3. Hessian measure

对非光滑 admissible 函数，也可以定义 Hessian measure。若 $u_j\to u$，并且 $u_j$ 是光滑 admissible，那么在合适条件下

$$S_k(D^2u_j)\,dx \rightharpoonup \mu_k[u].$$

这把方程扩展到 viscosity/pluripotential 风格的弱解框架。它类似 Monge-Ampere measure，但 $k$-Hessian 的 cone 结构更复杂。

4. Wolff potential

点态估计中会出现 Wolff potential：

$$W_{\alpha,p}^\mu(x)=\int_0^\infty\left(\frac{\mu(B(x,r))}{r^{n-\alpha p}}\right)^{1/(p-1)}\frac{dr}{r}.$$

它描述右端测度在不同尺度上的集中。对 $k$-Hessian 方程，解的上下界可以用相应的 Wolff potential 控制。这个形式可以从 scaling 看出：非线性阶数决定了势函数的指数。

5. 与几何曲率方程的联系

$k$-curvature 方程通常把 hypersurface 的第 $k$ 个曲率函数固定下来。解析上它和 Hessian 方程共享 symmetric polynomial、admissible cone 和 fully nonlinear ellipticity。几何问题中的正则性，往往依赖同一套先验估计和弱测度理论。

2017年11月14日
Vinogradov mean value theorem：矩估计、Weyl sums 与 decoupling

旧博客原文

原题：Note on Vinogradov main theorem

1.Introduction

Question:
Vinogradov mean value
Let $k,s\in \mathbb N$ , $x\in R^k$ .

$J_{s,k}(N)=|\{(n_1,...,n_s,n_{s+1},...,n_{2s})|n_1^j+...+n_s^j=n_{s+1}^j+...+n_{2s}^j) \forall 1\leq j\leq k,1\leq n_i\leq N(1\leq i\leq s) \}|$
How to estimate $J_{s,k}(N)$ ?

We assume $f_{k}(x,N)=\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)$ , then by following clear calculate:

$\int_{[0,1]^k}|f_k(x,N)|^{2s}dx_1...dx_k =\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_k &=\int_{[0,1]^k}\sum_{1\leq n_1,...,n_{2s}\leq N}e^{2\pi i[(n_1+...+n_{2s})x_1+...+(n_1^k+...+n_{2s}^k)x_k-(n_{s+1}+...+n_{2s})x_1-...-(n_{s+1}^k+...+n_{2s}^k)x_k]} &=|\{(n_1,...,n_{2s})| n_1^j+...+n_s^j=n_{s+1}^j+...+n_{2s}^j,\forall 1\leq j\leq k\}|$

we have:
$J_{s,k}(N)=\int_{[0,1]^k}|f_k(x,N)|^{2s}dx_1...dx_k$
main conjecture:
$\forall \epsilon >0$ ,we have:
$J_{s,k}(N)<<N^{\epsilon}(N^s+N^{2s-\frac{1}{2}k(k+1)})$

theorem(Bourgain-Demeter-Guth)
Main conjecture hold in general.

2.Application

We have following directly application:

1.Waring problem

2.Bound Weyl sums.\

3.Zero-free region for Riemann-zeta function.

3.Relate to the decoupling theorem

Now we discuss the decoupling theorem. This theorem describe the phenomenon when we are considering the “expension” operator $E_{[0,1]}(g)$ cut off $E_{[0,1]}(g)$ into a lot of small boxes $E_{J}(g)$ , then the $L_{d(d+1)})$ norms of the operator could be bounded very well, in fact it is near orthonagonal.

[B-D-G]
Let $d\geq 2$ ,$0<\delta\leq 1$. Then for each ball $B\subset R^d$ of radious at least $\delta^{-d}$ .
$||E_{[0,1]}g||_{L^{d(d+1)}(w_B)}<< \delta^{-\epsilon}(\sum_{J\subset [0,1],|J|=\delta}||E_Jg||^2_{L^{d(d+1)(w_B)}})^{\frac{1}{2}}$
( $J$ runs over a partition of $[0,1]$ in $\delta$ -intervals)

Discretized version:
Now we discuss the discretization of decoupling type result. We could establish a relationship between the decoupling theorem and Vinogradov mean theorem. look at the sum:
$\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_n$
This could be view as a $2s$ norm of a constant function $h=1$ , with a lebergue measure $d\sigma$ on curve $\Gamma=\{(t,t^2,...,t^d):0\leq t\leq 1\}$ . this curve $\Gamma$ could be view as a canonical curve with non-vanish guess curvature.
$||\widehat {hd\sigma}||_{2s}^{2s}=\int_{R^{k}}|\int_{\Gamma}h(t,t^2...,t^n)e(tx_1+...+t^kx_k)|^{2s}d\sigma$

this is very similar with the restriction theorem:

[restriction theorem]
let $\Gamma$ be $n-1$ dimension parabolic in $R^n$ , then guess curvature of $\Gamma$ is non-vanish. $\sigma$ is a natural induced lebergue measure on $\Gamma$ , we have, for suitable exponents $p,p'$ come from rescaling arument.
$||\widehat{gd\sigma}||_{p'}\lesssim ||g||_p$

So it seems like these are the same thing, but unfortunately they are not,there are two things distinct them:
1.the density is defferent, it is a discrete sum in:
$\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_n$
but a continue integral in:
$||\widehat {hd\sigma}||_{2s}^{2s}=\int_{R^k}|\int_{\Gamma}h(t,t^2...,t^n)e(tx_1+...+t^kx_k)|^{2s}d\sigma$
so we need to construct a rescaling way to make the discretization one coverage to the continue one.a suitable fexiable function seems like $(w_B,B_{r}(c_B)),w_B(x)=(1-\frac{|x-c_B|}{R})^{-100k}$
2.there

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 mean value theorem 控制 Weyl sums 的高阶矩，是 Waring problem、指数和估计和 zeta 函数零点区域中的基础工具。Bourgain-Demeter-Guth 用 decoupling theorem 证明了主猜想。

Vinogradov mean value theorem 把 Weyl sums 的高阶矩与 moment curve decoupling 联系起来。

1. Mean value

设

$$S(\alpha)=\sum_{n\le N}e(\alpha_1n+\alpha_2n^2+\cdots+\alpha_kn^k).$$

Vinogradov mean value 研究

$$J_{s,k}(N)=\int_{[0,1]^k}|S(\alpha)|^{2s}\,d\alpha.$$

它也等于某个 Diophantine system 解的个数。

2. 主猜想

主猜想断言

$$J_{s,k}(N)\lesssim_\varepsilon N^\varepsilon\left(N^s+N^{2s-k(k+1)/2}\right).$$

两个项分别对应 diagonal solutions 和维数计数给出的主项。

3. 应用

这个估计直接用于 Waring problem，也给出 Weyl sums 的强上界。通过指数和控制，可以进一步进入 zeta 函数零点区域和等分布问题。

4. Decoupling 视角

考虑 moment curve

$$\gamma(t)=(t,t^2,\ldots,t^k).$$

decoupling theorem 描述 extension operator 在小区间分解后的 $L^p$ 几乎正交性。离散化后，它与 Vinogradov mean value theorem 精确相连。

5. 思想总结

Vinogradov mean value 把数论中的方程计数、调和分析中的 Fourier extension、以及几何中的曲率结构放到同一个问题里。这是现代解析数论和 decoupling 理论交汇的代表。

2017年11月13日
从 linear 到 multilinear：Cauchy-Schwarz、分解与复杂度下降

旧博客原文

原题：Linear to Multi-linear

The technique that transform a problem which is in a linear setting to a multilinear setting is very powerful.

such like:

1.The renormalization technique in complex dynamic system, and the generalization

this is mainly the Ostrowoski representation,and something else.

2.Fouriour analysis

this can be view when it is difficult to investigate a quality about a function $f$ , it is always easier to take charge with some some part of $f$ , in this case is given by $\hat f(\xi),\xi \in R$ or $\hat f(k),k\in Z$ like cut $f$ into a lot of small parts,deal with every part and use some inequality(always the triangle inequality or similar thing) to glue it into a whole estimate of the quantity of $f$ .

3.Multi-scales theory

this is used in the improve of Minkowski dimension of 3-dim kakeya set by Katz-Tao.

4.The proof of Bourgain-Sarnak-Ziegler theorem

Theorem(B-S-Z). Let $F : N \to C$ with $|F| \leq 1$ and let $\nu$ be a multiplicative function with $|\nu| \leq 1$ . Let $\tau > 0$ be a small parameter and assume that for all primes $p_1, p_2 \leq e^{1/\tau}$ , $p_1 \neq p_2$ , we have that for $M$ large enough

$|\sum_{m\leq M} F(p_1m)\overline {F(p_2m)}| \leq \tau M$ .

Then for N large enough

$|\sum_{m\leq M} \nu(n)F(n) | \leq 2 \sqrt{\tau log(\frac{1}{\tau})}M$ .

this theorem is not difficult to prove by bilinear method and Cauchy-Schwarz,you can see the detail in https://arxiv.org/abs/1110.0992v1.
According to this theorem,to get a good approximation of $\sum_{1\leq k\leq x}\mu(k)f(k)$ we use need a good approximation on $\sum_{1\leq k\leq x}f(p_1x)\overline {f(p_2x)}$ .this will be much easier.but for the RHS $f(x)$ is very complicated so I do not have a non-trivial estimate for $\sum_{1\leq k\leq x}f(p_1x)\overline {f(p_2x)}$ until now.

5.The multiplier restriction theorem(Tao)

6.Some special construction in additive Combitriocs

Such like when we want to consider some set $A\subset Z_1$ satisfied $\frac{|A-A|}{|A+A|}>>1$ it is convenient to consider in a high dimensional linear space $Z^N$ rather than in $Z$ .

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。

很多问题直接估计一个线性量很难，但把它平方、分解或转成多线性形式后，结构反而清楚。这就是从 linear 到 multilinear 的基本哲学。

从 linear 到 multilinear 的基本动作是用 Cauchy-Schwarz 或分解降低复杂度，同时保留主要信息。

1. Cauchy-Schwarz 的真正作用

Cauchy-Schwarz 不只是让估计变粗。好的使用方式是：降低复杂度，同时几乎不损失主要信息。例如

$$\left|\sum_n a_n b_n\right|^2\le \left(\sum_n|a_n|^2\right)\left(\sum_n|b_n|^2\right).$$

如果右边的两个平方和更容易理解，这一步就是有效的。

2. Fourier 分解

在调和分析中，一个函数被切成频率块：

$$f=\sum_\theta f_\theta.$$

线性估计可能无法直接控制 $\sum_\theta f_\theta$，但双线性或多线性估计能利用不同 $\theta$ 之间的 transversal 结构。

3. Multiscale 理论

Kakeya 和 restriction 中，多尺度分解把一个集合或函数切成不同尺度上的 pieces。每一层可能只给出局部信息，但层与层之间的递归关系能产生全局维数或范数估计。

4. BSZ 准则

Bourgain-Sarnak-Ziegler 准则就是线性到双线性的典型例子。要证明

$$\sum_{n\le N}\mu(n)a_n=o(N),$$

可以转而控制

$$\sum_{n\le N}a_{pn}\overline{a_{qn}}$$

对不同素数 $p\ne q$ 的相关。这把乘法函数的线性相关转成序列自身的双线性相关。

5. 方法的边界

多线性化不是免费午餐。右边的量可能更复杂，分解后还要重新 glue 回整体估计。好的多线性方法，总是在“降低复杂度”和“保持信息”之间找到平衡。

2017年10月30日
Gromov-Hausdorff 距离笔记：不同维球面之间能多近？

旧博客原文

原题：How to compute the Gromov-Hausdorff distance between spheres $latex S_n$ and $latex S_m$?

There is the question, because when we consider the Gromov-Hausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $\mathbb{S}_n \to \mathbb{R}^{n+1}$ . Is it possible for us to compute the Gromov-Hausdorff distance $d_{G-H}(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$ , $m\neq n$ ?

For example if we want to calculate $d_{G-H}(\mathbb{S}_2,\mathbb{S}_3)=\inf_{M,f,g}d_{M}(\mathbb{S}_2,\mathbb{S}_3)$ , where $M$ ranges over all possible metric space and $f:\mathbb{S}_2\to M$ and $g:\mathbb{S}_3\to M$ range over all possible isometric (distance-preserving) embeddings.

At least we can embed $\mathbb{S}_2$ , $\mathbb{S}_3$ into $\mathbb{R}^3$ in a canonical way. This will lead to a upper bound: $d_{G-H}(\mathbb{S}_2,\mathbb{S}_3)\leq \sqrt{2}$ . And in general case we have $d_{G-H}(\mathbb{S}_m,\mathbb{S}_n)\leq d_{G-H}(point,S_m)+d_{G-H}(point,S_n)\leq 2,\forall 0\leq n\leq m$ . But it is difficult to get a lower bound control for me. Because we need to take the inf in all possible metric spaces $M$ . Especially I conjecture $d_{G-H}(\mathbb{S}_m,\mathbb{S}_n)\geq \lambda_{m,n}\frac{m-n}{m},\forall 0\leq n\leq m$ , where $\liminf_{m,n\to \infty}\lambda_{m,n}>0$ .

I only know the knowledge of Gromov-Hausdorff from Peterson’s Riemann Geometry. Unfortunately there is not enough information to compute the Gromov-Hausdorff distance, so this problem may be very stupid, I will appreciate any pointer.

And we know for the case $S_n,S_m$ , if $n,m$ is very near to each other,then the two space should be more near, and there is a canonical embed $S_0\subset S_1 \subset S_2 ....\subset S_n \subset ...$ . So it is natural to conjecture if $m,n$ is very near then the distance $d_{G-H}(S_n,S_m)$ is very small. I have a very rough strategy to prove the conjecture, that is inspired by the Nash embedding theorem. I just mean if we consider the problem in this frame $d_{G-H}(S_n,S_m)=\inf_{M,g,f}(d_M(f(S_n),g(S_m)))$ then the difficult is the deformation space of $M,g,f$ is too large. so the first step is to establish a regular lemma, to prove the function $d_M(f(S_n),g(S_m))$ is continues under the small perbutation of $M$ and reduced to the situation of space $M,g,f$ with very nice regularity. the second part is to embed $M$ to a big euclid space $R^N$ as subspace, and the embedding stay the length of geodesic.locally this is determine by a group of pde: $u_i(x)u_j(x)=g_{ij}(x)$ ,at least in the cut locus.but there should be some critical point,and I do not know how to deal with them.the third,i.e. the last step is to calculate $d_{G-H}(S_n,S_m)$ in the very some deformation space $M,f,g$ .

@Mark Sapir,Appreciate for help!I am reading the article you point out,it seems this article mainly focus on investigating the Gromov-Hausdorff limit space of a sequence of hyperbolic group equipped with modified G-H metric defined in 2.A with some special condition to ensure the limit space exists.and take a sequences corvarage to the limit space,the hyperbolic property and some other thing is stayed by the process of take limit.
@Mark Sapir,So it is natural for us to investigate the original space by some information from the limit space.there is a series of bi-product state in 3.B.but I do not see where the author exactly calculate some groom-hausdorff distance of two different space,may you point out it?appreciate again!
@MarianoSuárez-Álvarez,Corrected, thanks.

Y:
I fixed numerous typos. In particular, you should use spacing after each punctuation mark; capitals to begin sentences and names.

H:
Thank you very much for helping me to correct the mistakes! I will know how to write in a correct style.

Y:
23.1k
Your conjecture would imply that the GH distance is unbounded. But it’s clearly bounded, since the GH distance of any sphere to a point is equal to 2 (when the sphere is endowed with the restriction of Euclidean distance, as you seem to assume, or $\pi$ when endowed with geodesic distance) and hence the GH distance between any two spheres is $\le 4$ .

H:
73
You are right,In fact if we use the canonical embed, then we can get $d_{G-H}(S_n,S_m)\leq 2$ by another equivalent definition of GH distance.I confuse the geometry picture of the pairs $T_n,S_n$ with the pairs $S_n,S_m$ ,for $S_n,S_m$ case,I thick the seems correct conjecture will be $d_{G-H}(S_n,S_m)\sim \frac{m-n}{m},0\leq n\leq m,m,n\to \infty$ .

Y:
16:37
Clearly from standard embeddings we get $d_{GH}(S_n,S_m)\le\sqrt{2}$ for all $n,m\ge 0$ . Would it be reasonable to simply conjecture that it’s an equality whenever $n\neq m$ ?

H:
73
Yeah, you are right, $d_{G-H}(S_n,S_m)\leq \sqrt{2}$ for all $n,m\geq 0$ .I find the interesting problem when I want to find a toy model of a kind of problem,roughly speaking is to investigate a map $f:X\to Y$ from low-dimensions space $X$ to high-dimension space $Y$ stay some affine structure of the low-dimension space $X$ . This structure could have some control by the distance function on the low-dimension space, so if we can get some control on the variation of the Energy of distance function, this will share some line on the original problem I consider.
And we know for the case $S_n,S_m$ , if $n,m$ is very near to each other,then the two space should be more near, and there is a canonical embed $S_0\subset S_1 \subset S_2 ....\subset S_n \subset ...$ . So it is natural to conjecture if $m,n$ is very near then the distance $d_{G-H}(S_n,S_m)$ is very small. .
I have a very rough strategy to prove the conjecture, that is inspired by the Nash embedding theorem. I just mean if we consider the problem in this frame $d_{G-H}(S_n,S_m)=\inf_{M,g,f}(d_M(f(S_n),g(S_m)))$ then the difficult is the deformation space of $M,g,f$ is too large. so the first step is to establish a regular lemma, to prove the function $d_M(f(S_n),g(S_m))$ is continues under the small perbutation of $M$ and reduced to the situation of space $M,g,f$ with very nice regularity.
The second part is to embed $M$ to a big euclid space $R^N$ as subspace, and the embedding stay the length of geodesic.locally this is determine by a group of pde: $u_i(x)u_j(x)=g_{ij}(x)$ ,at least in the cut locus.but there should be some critical point,and I do not know how to deal with them.the third,i.e. the last step is to calculate $d_{G-H}(S_n,S_m)$ in the very some deformation space $M,f,g$ .
I need come back to explain why we expect the groom-hausdorff distance $d_{G-H}(S_d,S_m),0\leq n\leq m$ should be much small than $\sqrt 2$ when $frac{n}{m}$ is small.
Let consider a toy model of the problem,in a graph model,i.e. now we do not consider to take the Infimum in all space but in discrete space endow with metric. this can be view as a complete graph equipped metric, i.e. $M=\{(G,d_G)\}$ . So there is also some space very like $S_n$ ,$S_m$ in the Euclid space, Let remark them as $G_{S_n},G_{S_m}$ .
, oberseve that $(G,d_G)\in M$ then $(G,\hat d_{G})\in M$ , $\hat d_{G}$ is a scaling of $d_G$ so it is natural to consider a cut off of $M$ ,called $M_{\lambda}$ which is just a subset of $M$ and satisfied if $(G,d_G)\in M_{\lambda}$ ,then $\inf_{x\neq y}d_{G}(x,y)\geq d$ .
Now,in the space $M_{\lambda}$ let us consider a Distance distribution: $\mu_G((a,b))=\frac{\#\{x,y\in G|a<d_G(x,y)<b\}}{\#G\times G}$ . Then this distribution will give us some information of the distance of the two different set $G_1,G_2$ in $M_{\lambda}$ .
(removed)
Now,in the space $M_{\lambda}$ let us consider a Distance distribution: $\mu_G((a,b))=\frac{\#\{x,y\in G|a<d_G(x,y)<b\}}{\#G\times G}$ . Then this distribution will give us some information of the distance of the two different set $G_1,G_2$ in $M_{\lambda}$ .
and obviously we will see that if $n,m$ is close,then the distribution of fuzzy approximation $G_{S_n},G_{S_m}$ is near, and the reverse is also true. I think this can explain why the conjecture $d_{G-H}(S_n,S_m) =O(\frac{m-n}{m})$ may be right.

by the way,it is a very good exercise to proof $d_{G-H}(S_n,S_0)=1,\forall n\in N^*$ .

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。

问 $S^n$ 和 $S^m$ 的 Gromov-Hausdorff 距离，第一步必须先固定 metric。球面可以带内蕴测地距离，也可以带作为欧氏空间子集的 chordal distance；不同选择会给出不同数值。真正稳定的对象不是某个嵌入，而是所有可能 correspondences 的 distortion。

比较不同维球面时，标准嵌入只给出上界；真正的 GH 距离要在所有 correspondences 上取最优。

1. 定义提醒

紧 metric spaces $X,Y$ 的 Gromov-Hausdorff 距离可以用 correspondences 表示：

$$d_{GH}(X,Y)=\frac12\inf_R \operatorname{dis}(R),$$

其中 $R\subset X\times Y$ 是 correspondence，distortion 定义为

$$\operatorname{dis}(R)=\sup_{(x,y),(x’,y’)\in R}
\bigl|d_X(x,x’)-d_Y(y,y’)\bigr|.$$

这一定义强调：我们不是只比较两个空间在某个欧氏空间中的位置，而是允许把它们同时嵌入任意 metric space，再取最优比较。

2. 先用标准嵌入给上界

若 $n
$$d_{GH}(S^n,S^m)\le \sqrt2.$$

若用 geodesic metric，对应上界是 $\pi/2$。这只是上界，不自动说明最优。

3. 为什么不能猜它随维数增长

一个容易犯的错误是认为维数差越大，距离越大。但 GH 距离由直径控制。对任意紧空间 $X$，

$$d_{GH}(X,\{\ast\})=\frac12\operatorname{diam}(X).$$

因此两单位球面之间的 GH 距离总是被一个统一常数控制，不可能随着 $m-n$ 无界增长。维数差体现为拓扑和覆盖数的差异，但 GH 距离本身仍然是 metric 层面的量。

4. 下界为什么难

要证明 equator 嵌入给出的上界是最优，需要排除所有更聪明的 correspondences。这通常要找一个 metric invariant，例如 covering number、packing number、waist phenomenon 或同调信息，说明低维球面无法在小 distortion 下模拟高维球面。

例如，如果 $S^m$ 中有许多两两相距较远的点，而 $S^n$ 在同样尺度下容纳不了这么多点，就可以得到下界。这类 argument 本质上是把维数信息转化成 packing 数据。

5. 一个合理的 toy problem

这个问题真正有意思的地方在于，它是“低维空间如何嵌入高维空间并保留距离结构”的 toy model。若把球面换成带变形 metric 的流形，还会涉及 Nash embedding、cut locus、距离函数能量的变化以及正则性问题。

所以比较 $S^n$ 和 $S^m$ 不只是为了得到一个数值；它逼迫我们区分三件事：具体嵌入给出的 Hausdorff 上界、抽象 GH 距离的最优 correspondence、以及维数信息如何通过 metric invariant 被读出来。

2017年10月28日
k-curvature 方程的正则性：连续性方法、屏障函数与边界估计

旧博客原文

原题：Regularity of k-curvature equation

this is a note after reading the article”” of Cafferalli.

in his article,a large type of fully nonlinear elliptic equation has been established.in particular,including the k-curvature equation.and use the continue method,we just need to establish a ingredient estimate, $C^2$ estimate in the interior and $C^2$ estimate near the boundary.we establish these estimate step by step,base on construct special flexible function and use the maximum principle to establish the first and second estimate,for the $C^2$ estimate near the boundary we need to investigate the influence of permutation on the boundary carefully.

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。

$k$-curvature 方程属于完全非线性椭圆方程。典型形式可以写成

$$F(D^2u,Du,u,x)=\sigma_k(\kappa[u])=\psi(x),$$

其中 $\sigma_k$ 是主曲率的第 $k$ 个初等对称函数。Caffarelli-Nirenberg-Spruck 一类理论的核心是：在合适凸性锥中建立先验估计，然后用连续性方法得到解。

k-curvature 方程的正则性证明依靠连续性方法、屏障函数、最大值原理和边界二阶估计。

1. 连续性方法

把目标方程嵌入一族方程 $F_t(u)=0$。若 $t=0$ 可解，开性来自线性化算子的可逆性，闭性则依赖统一先验估计。于是关键变成：

$$\|u\|_{C^{2,\alpha}}\le C.$$

2. 内部估计

$C^0$ 和 $C^1$ 估计通常来自最大值原理与几何屏障。二阶估计更微妙，需要对最大特征值构造辅助函数，并利用方程的凹性抵消坏项。

3. 边界估计

边界附近最难的是混合二阶导数和法向二阶导数。这里要仔细利用边界曲率、可容许锥条件，以及对称函数 $\sigma_k$ 在坐标置换下的结构。

4. 正则性升级

一旦得到一致椭圆性和 $C^2$ 控制，可以通过 Evans-Krylov 定理得到 $C^{2,\alpha}$，再用 Schauder theory 提升到更高正则性。整个证明的脊梁是：屏障函数给低阶控制，最大值原理给二阶控制，椭圆正则性完成升级。

2017年10月24日
Wolff 的 Kakeya maximal estimate：X-ray 估计与三维障碍

旧博客原文

原题：Kakeya conjecture (Tomas Wolff 1995)

There is a main obstacle to improve the kakeya conjecture,remain in dimension 3,and the result established by Tomas Wolff in 1995 is almost the best result in $R^3$ even until now.the result establish by Katz and Tao can be view as a corollary of Wolff’s X-ray estimate.

For $f\in L_{loc}^1(R^d)$ ,for $0<\delta<1$ :

$f_{\delta}^*:P^{d-1}\longrightarrow R$ . $f_{\delta}^*(e)=\sup_{T}\frac{1}{|T|}\int_{T}|f|$ .

$T$ is varise in all cylinders with length 1.radius $\delta$ .axis in the $e$ direction.

$f_{\delta}^{**}:R^d\longrightarrow R$ . $f^{**}_{\delta}(x)=sup_{T}\frac{1}{|T|}\int_{T}|f|$ .

$T$ varise in cylinders contains x,length 1,radius $\delta$ .

Keeping this two maximal function in mind,we give the statement of the Kakeya maximal function conjecture:

$||M_{\delta}f||_d\leq C_{\epsilon} \delta^{-\epsilon}||f||_d$

Where $M_{\delta}=f_{\delta}^*$ or $M_{\delta}=f_{\delta}^{**}$ .

Because we have the obviously $1-\infty$ estimate:

$||f^*_{\delta}||_{\infty}\leq\frac{||f||_1}{|T|}=\delta^{1-d}||f||_1$ .

$||f^{**}_{\delta}||_{\infty}\leq\frac{||f||_1}{|T|}=\delta^{1-d}||f||_1$ .

So by the Riesz-Thorin interpolation we have:

$||M_{\delta}f||_{q}\leq C_{\epsilon}\delta^{-(\frac{d}{p}-1+\epsilon)}||f||_p$ . (*)

for $1\leq p\leq d,q\leq(d-1)p'$ .the task is establish (*) for $(p,q)$ as large as posible in the range.

for the 2 dimension case,the result is well know.the key estimate is:

$\sum_{j}|T_i\cap T_j|\leq log(\frac{1}{\delta})|T_i|$

for $d\geq 3$ case,the main result of Wolff is:

$||M_{\delta}f||_q\leq C_{\epsilon}\delta^{-(\frac{d}{p}-1+\epsilon)}||f||_p$

hold for $p=\frac{d+2}{2}.q=(d-1)p'$ . $M_{\delta}=f_{\delta}^*$ or $f_{\delta}^{**}$ .

Now we sketch the proof.

prove $f_{\delta}^*,f_{\delta}^{**}$ cases together.

We can make some reduction:

the first one is we can assume the sup of $f$ is in a fix compact set.

the second is instead of consider $f_{\delta}^{**}$ ,we can consider $f_{\delta}^{***}(x)=\sup_{T}\frac{1}{|T|}\int_T|f|$ .

where $T$ varies in all cylinder with radius $\delta$ ,length 1,axis $\frac{\pi}{100}$ with a fix direction.

the first reduction is obvious(why?)

the second reduction rely on a observe:

$||f_{\delta}^{***}||_q\leq A(\delta)||f||_p$ $\Longrightarrow$ $||f_{\delta}^{**}||_q\leq CA(\delta)||f|_p$

this is just finite cover by rotation of the coordinate and triangle inequality.

now we begin to establish a frame and put the two situations $f_{\delta}^*,f_{\delta}^{***}$ into it.

Let $M(d,1)$ be all line in $R^d$ .

then $M(d,1)=R^d\times S^{d-1}/\sim$ is a $2d-2$ dim manifold.

$M(d,1)\longrightarrow P^{d-1}$

$l \longrightarrow e_l$

$e_l$ is the line parallel to $l$ .and the middle point is original.

$dist(l_1,l_2)\sim \theta(l_1,l_2)+d_{mis}(l_1,l_2)$ .

Wolf axiom:

$(A,d)$ metric space.

$\mu(D(\alpha,\delta)) \sim \delta^m$ . $\alpha\in A$ . $\delta \leq diam(A)$ .

for certain $m\in R^+$ .

$\forall \alpha\in A$ . $F_{\alpha} \subset M(d,1)$ is given.and $\bar{\cup_{\alpha}F_{\alpha}}$ is compact.

$d(\alpha,\beta)\lesssim inf_{l\in F_{\alpha};m\in F_{\beta}}dist(l,m)$ for all $\alpha,\beta \in A$ .

If $f:R^d\longrightarrow R$ then we define $M_{\delta}f:A\longrightarrow R$ by

$M_{\delta}f(\alpha)=\sup_{l\in F(\alpha)}\frac{1}{|T_{l}^{\delta}|}\int_{|T^{\delta}_l|}|f|$ .

Property (**):

If $l_0\in \cup_{\alpha F_{\alpha}}$ . $\Pi$ is a 2-plane.containing $l_0$ and if $\sigma \geq \delta$ and if $\{\alpha_j\}_{j=0}^N$ is a $\delta-seperated$ subset of $A$ and for each j,there is $l_j\in F_{\alpha_j}$ with $dist(l_j,M(\Pi,l))<\delta$ and $dist(l,l_0)<\sigma$ .then

$N\leq \frac{C\sigma}{\delta}$

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。

Wolff 1995 年的 Kakeya maximal estimate 是三维 Kakeya 理论中的经典结果。它把 tubes 的重叠问题转化为 maximal function 和 X-ray transform 的估计。

Wolff 的 Kakeya maximal estimate 通过 X-ray 型平均和 incidence 几何控制 tubes 的重叠。

1. Kakeya maximal function

给定方向 $\omega$，考虑沿 $\omega$ 方向、半径 $\delta$、长度 $1$ 的 tube。Kakeya maximal function 粗略写为

$$f_\delta^\ast(\omega)=\sup_{T\parallel\omega}\frac1{|T|}\int_T |f(x)|\,dx.$$

猜想要求在尽可能大的 $p$ 范围内控制 $\|f_\delta^\ast\|_{L^q(S^{n-1})}$。

2. 平凡估计与插值

最直接的 $L^1$ 或 $L^\infty$ 估计通常太弱。通过 Riesz-Thorin interpolation 可以得到一些中间结果，但要接近 Kakeya 猜想，需要利用 tubes 的几何排列。

3. X-ray transform

X-ray transform 沿直线积分函数。Kakeya maximal function 可以看作对有限厚度 tubes 的 X-ray 型平均。Wolff 的估计用几何 incidence 控制这些平均的重叠。

4. 归约步骤

证明中可以把 tubes 限制在固定紧区域，并把方向分成有限个坐标 patch。这样 maximal function 可由有限方向族上的 tube averages 控制。

5. 三维障碍

三维最大的困难是 tubes 可以形成复杂 ruled surface 或 hairbrush 结构，普通 pairwise intersection 估计不够。Wolff 的结果抓住了这类结构的第一层几何约束，为 Katz-Tao 的后续改进提供了基础。

2017年10月24日
Julia set 的 Hausdorff 维数能否等于 2？

旧博客原文

原题：Julia set

There is a major open problem:

Is there a polynomial $f(z)$ such that the julia set $T(J)$ of map $T:z\longrightarrow f(z)$ satisfied the hausdorff dimension of J equal to 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。

复动力系统中的 Julia set 是迭代多项式

$$f^{\circ n}(z)$$

稳定性发生断裂的边界。一个自然问题是：是否存在多项式 $f$，使得 Julia set 的 Hausdorff 维数等于 $2$？也就是说，它虽然可能没有面积，却在维数意义上填满平面。

Julia set 的满 Hausdorff 维数问题连接复动力系统、分形几何和非一致双曲性。

1. Fatou set 与 Julia set

Fatou set 是迭代族正常的区域，Julia set 是它的补集。对 $f(z)=z^2+c$ 这样的二次多项式，Julia set 的形状会随参数 $c$ 发生剧烈变化。

2. 维数为什么可能接近 2

双曲多项式的 Julia set 往往有清晰的膨胀结构，维数可以通过压力函数计算。若系统接近抛物或临界退化，膨胀变弱，轨道在小尺度上停留更久，维数可能被推高。

3. 等于 2 的含义

$\dim_H J(f)=2$ 不等于说 $J(f)$ 有正面积。Hausdorff 维数为 $2$ 只说明它在任意小尺度上足够复杂；Lebesgue 面积是否为正是更强的问题。

4. 研究路线

一种思路是构造参数序列，使 Julia set 的维数趋近 $2$，再分析极限是否保持多项式动力系统的性质。另一种思路是研究临界轨道的回归速度和尺度几何，寻找能够制造满维数的非一致双曲机制。

2017年10月21日
用互不相交闭区间覆盖半开区间：Ostrowski 表示与有效均匀分布

旧博客原文

原题：Covering a non-closed interval by disjoint closed intervals

this note will talk about the Ostrowski representation and approximation by continue fraction.

As well-known,by the Weyl criterion, $\{n\alpha\}$ is uniformly distribution in $[0,1]$ iff $\alpha\in R-Q$ .

i.e. we have: $\forall 0\leq a\leq b\leq 1$ ,we have:

$\lim_{N\to \infty}|\{1\leq n\leq N|\{n\alpha\}\in [a,b]\}|=(b-a)N+o(N)$ .

but this will not give the effective version.i.e. we do not the the more information about the decay of $o(N)$ .

we will give a approach of effective version of $\alpha$ with smooth condition by give another proof of the uniformly distribution (in fact to to decomposition the interval $[a,b]$ in to a finite sums of special intervals).and get the result:

$D_N=\int_{M}D_N(\theta)d\mu=\int_Msup_{0<a<b<1}|\sum_{n=1}^{N}\chi_{(a,b)}(\{\theta n \})-N(b-a)|d\mu\sim O(log N)$

if the term in the continuous fraction of $\alpha$ have a up bound.this is so called $\alpha$ is smooth.

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。

讨论无理旋转 $x\mapsto x+\alpha$ 的均匀分布时，Weyl criterion 告诉我们

$$\frac1N\sum_{n\le N}e(k n\alpha)\to0,\qquad k\ne0.$$

但这个判别法本身并不给出非常几何的有效误差。若想知道轨道落入某个区间 $I$ 的次数和 $N|I|$ 相差多少，就需要把区间和时间长度一起分解得更细。

Ostrowski 表示把轨道长度分解成连分数分母尺度，半开区间也可以拆成有限个互不相交的闭区间来控制端点误差。

1. 半开区间为什么麻烦

区间若不是闭的，端点会造成计数上的小麻烦；但在动力系统里，端点通常只贡献有限误差。因此可以把半开区间分解成有限个互不相交的闭区间，再把端点误差单独处理。

2. 连分数与最佳逼近

设 $\alpha=[a_0;a_1,a_2,\ldots]$，其收敛分母为 $q_j$。最佳逼近性质说明，长度约为 $\|q_j\alpha\|$ 的小区间正好对应旋转轨道的自然尺度。

Ostrowski 表示把整数 $N$ 写成

$$N=\sum_j b_j q_j,$$

其中系数 $b_j$ 由连分数项控制。于是前 $N$ 次轨道可以分成若干个以 $q_j$ 为长度的块。

3. 有效均匀分布

若 $\alpha$ 的连分数项有一致上界，也就是 bounded type，那么每个尺度的坏误差都不会积累得太快。这样可以得到 Denjoy-Koksma 型估计：

$$\left|\sum_{n
这就是原来想要的“smooth”或有效版本：不仅知道趋于均匀，还知道偏差怎样增长。

2017年10月20日