Relationship of relationship

所谓的Minkowski猜想，就是我们有一个$SL(n,R)$中的一个离散子群。在这里，$SL(n,R)$是所有$n$阶可逆矩阵构成的群，这些矩阵的行列式是1。我对这个猜想感兴趣，是因为它与Lonely Runner Conjecture有一些相似之处。在Lonely Runner Conjecture中，我们也是在高维空间中进行研究，但我们的研究对象是高维空间中的一维子空间。我们希望在这个一维子空间上进行优化问题的研究，而这个优化问题涉及到的是对底空间$S^1$的遍历。 然而，在Minkowski猜想中，我们的底空间并不大，它是$R^n$。我们需要研究的格点空间相当大，它可以看作是$SL(n,R)$模一个离散子群。这个空间的维度大约是$n$的平方，所以我们可以将其视为一个非常大的空间。在这个大空间中，我们需要找到所有的格点，在这个大约是$n$平方维度的空间中，找到对应的最大值。这个最大值实际上是指空间中某个点到最近的若干个顶点的距离的乘积的最大值。我们的目标就是找到这样的一个点，使得这个乘积最大。 我们是要遍历所有的格点，格点的空间大概是$n$平方维度，这个空间面遍历，所有的目标是找…

Although the $L^1$ norm of the Dirichlet kernel and the Pólya-Vinogradov inequality belong to different areas of mathematics, they share some connections.  L1 norm of the Dirichlet kernel $$D_n(x)=\sum_{k=-n}^n e^{i k x}=\left(1+2 \sum_{k=1}^n \cos (k x)\right)=\frac{\sin ((n+1 / 2) x)}{\sin (x / 2)}$$ The Dirichlet kernel is mainly used to describe the pointwise relationship between a function f and its Fourier transform hat f. We have a famous conjecture, the Lusin conjecture, which roughly de…

In this blog, we mark some article that is important in the investigation of the equidistribution problem, each of them providing new inspirit to solve some particuler type of equidistribution problem. 1. Mixing, counting, and equidistribution in Lie groups The author is Alex Eskin, Curt McMullen This article was published in 1993. in Duke Math. J. 71(1): 181-209 (July 1993). DOI: 10.1215/S0012-7094-93-07108-6 Let $\Gamma \subset G=\operatorname{Aut}\left(\mathbb{H}^2\right)$ be a group of isome…

A projection operator is a linear mapping from a linear space X to X that then satisfies $p^2=p$ Then in a geometrical point of view, looking at the linear space, any element in $X$ mapping under $p$ only has two possibilities, one is $px=x$, and under the action of $p$ image of $x$ is himself. The second possibility, x under the action of $p$ is not itself but another element $y$, then $p$ in action, on $y$ should also be equal to $y$ because we know that $p^2x=px = y$ so $py=p^2x=px = y$. Well…

分析中很多问题都可以用取平均来解决，但是能用取平均解决的问题，本质上是trivial的。 Lonely Runner conjecture本质上是一个不能用取平均解决的问题,整个框架就不对, 可以看成一个粒子扩散的问题，到地方那看这个就可以看出一个例子扩散的问题，但是它是非常非常简单的情形，在这个情形下，整个全空间是$S_1$或者$Z_p$。

The Ramanujan-Nagell Theorem: Understanding the Proof

The pdf version is A crash introduction to BSD conjecture . We begin with the Weierstrass form of elliptic equation, i.e. look it as an embedding cubic curve in $latex {\mathop{\mathbb P}^2}&fg=000000$. Definition 1 (Weierstrass form) $latex {E \hookrightarrow \mathop{\mathbb P}^2 }&fg=000000$, In general the form is given by, $latex \displaystyle E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)&fg=000000$ If $latex {char F \neq 2,3}&fg=000000$, then, we have a much more simpe…

The pdf version is A crash introduction to BSD conjecture . We begin with the Weierstrass form of elliptic equation, i.e. look it as an embedding cubic curve in $ {\mathop{\mathbb P}^2} $. Definition 1 (Weierstrass form) $ {E \hookrightarrow \mathop{\mathbb P}^2 } $, In general the form is given by, $  E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)$ If $ {char F \neq 2,3} $, then, we have a much more simper form, $ y^2=x^3+ax+b, \Delta:=4a^3+27b^2\neq 0. \ \ \ \ \ (2) $ Remar…

A pdf version is Dirichlet hyperbola method. 1. Introduction Theorem 1 $\sum_{1\leq n\leq x}d(n)=\sum_{1\leq n\leq x}[\frac{x}{n}]=xlogx+(2\gamma-1) x+O(\sqrt{x}) \ \ \ \ \ (1)$ Remark 1 I thought this problem initial 5 years ago, cost me several days to find a answer, I definitely get something without the argument of Dirchlet hyperbola method and which is weaker but morally the same camparable with the result get by Dirichlet hyperbola method. Remark 2 How to get the formula: $\sum_{1\leq n\le…

I explain some general ideal in the theory of diophantine approximation, some of them is original by myself, begin with a toy model, then consider the application on folklore Swirsing-Schmidt conjecture. 目录 Contents 1. Dirichlet theorem, the toy model The very basic theorem in the theory of Diophantine approximation is the well known Dirichlet approximation theorem, the statement is following. Theorem 1 (Dirichlet theorem) for all ${\alpha}$ is a irrational number, we have infinity rational numb…