Relationship of relationship

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}…

The most important beakgrouth of analytic number theory is the new understanding of multiplication function on share interval, this result is established by Kaisa Matomäki & Maksym Radziwill. Two very young and intelligent superstars. The main theorem in them article is : Theorem(Matomaki,Radziwill) As soon as $latex H\to \infty$ when $latex x\to \infty$, one has: $latex \sum_{x\leq n\leq x+H}\lambda(n)= o(H)$ for almost all $latex x\sim X$ .   In my understanding of the result, the mai…

https://en.wikipedia.org/wiki/Transversality_(mathematics) This problem may be a embarrassed one, but I even could not prove it for the 1 dimensional case. Here is the problem: >**Question 1** $latex M$ is a compact $latex n$-dimensional smooth manifold in $latex R^{n+1}$, take a point $p\notin M$. prove there is always a line $latex l_p$ pass $latex p$ and $latex l_p\cap M\neq \emptyset$, and $latex l_p$ intersect transversally with $latex M$. You can naturally generated it to: >**Qusetio…

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&#8221…

Direchlet principle: $ \Omega \subset R^n$ is a compact set with $C^1$ boundary. then there exists unique solution $f$ satisfied $\Delta f=0$ in $\Omega$, $f=g$ on $ \partial \Omega$. Perron lifting and barrier function We know the standard approach of the Dirichlet principle is perron lifting and construction of barrier function on the boundary. The key point is if we define the variation energy $ E(u)=\int_{\Omega}|\nabla u|^2$, then it is easy to see for $ u_1,u_2$ is in perron set, $ E(sup (…

In 1911 year, when Weyl is a young mathematician specticlizing in integrable system and PDE, He proved the important result about the asystomztion of eigenvalues of Dirichelet problem in $latex \Omega\subset R^n$ is a compact domain;i.e. $latex N(\lambda)=(2\pi)^d Vol(\Omega)\lambda^{\frac{d}{2}}(1+o(1))$ Which in fact is a conjecture of *** in *** in published in 1910. This is definitely a very amazing achievement of mathematician, The realist meaning we can actually charge with the spectrum as…

Sarnak conjecture is a conjecture lie in the overlap of dynamic system and number theory. It is mainly focus on understanding the behavior of entropy zero dynamic system by look at the correlation of an observable and the Mobius function . We state it in a rigorous way: let $ (X,T)$ be a entropy zero topological dynamic system. Let Mobius function be defined as $ \mu(n)=(-1)^t$, where $ $ is the number of different primes occur in the decomposition of $ n$. Then for any continuous function $ f:X…

$latex M$ is the markov triple $latex (x,y,z)$: $latex x^2+y^2+z^2=xyz$ and $latex (x,y,x)\in \mathbb Z^3  \ \ \ \  (*)$. It is easy to see: $latex R_1: (x,y,z)\to (3yz-x,y,z)$. map markov triple to markov triple. This is also true for $latex R_2,R_3$. and the transform $latex R_1,R_2,R_3$ and permutation a classical result of markov claim that all solution of  (*) could be generated from $latex (1,1,1)$. I get a similar result for a similar algebraic equation 1 half years ago when consider a $l…

I am reading the article “ENTROPY THEORY OF GEODESIC FLOWS”. Now we focus on the upper semi-continuouty of the metric entropy map. The object we investigate is $latex (X,T,\mu)$, where $latex \mu$ is a $latex T-$invariant measure. The insight to make us interested to this kind of problem is a part of variational problem, something about the existence of certain object which combine a certain moduli space to make some quantity attain critical value(maximum or minimum). The most simple…