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Van der curpurt trick
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}…
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Multiplication function on short interval
The most important background of analytic number theory is the new understanding of the multiplication function on the shared interval. This result is established by Kaisa Matomäki & Maksym Radziwill, two very young and intelligent superstars. The main theorem in their article is: Theorem (Matomaki, Radziwill):As soon as $H \to \infty$ when $x \to \infty$, one has:$$\sum_{x\leq n\leq x+H}\lambda(n)= o(H)$$for almost all $x\sim X$. In my understanding of the result, the main strategy is: Step…
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transverse intersections
https://en.wikipedia.org/wiki/Transversality_(mathematics) Question 1: Let $M$ be a compact $n$-dimensional smooth manifold in $\mathbb{R}^{n+1}$, and take a point $p \notin M$. Prove that there is always a line $l_p$ passing through $p$ such that $l_p \cap M \neq \emptyset$, and $l_p$ intersects transversally with $M$. Question 2: Let $M$ be a compact $n$-dimensional smooth manifold in $\mathbb{R}^{n+m}$, and take a point $p \notin M$. Prove that for all $1\leq k\leq m$, there is always a hyper…
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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”…
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From periodic to quasi periodic
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A discret to continuous approach to the Dirichlet principle.
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 (…
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Weyl law
In the 1911 year, when Weyl was a young mathematician specializing in integrable systems and PDE, He proved the important result about the asystomztion of eigenvalues of Dirichlet problem in $latex \Omega\subset R^n$ is a compact domain;i.e. $N(\lambda)=(2\pi)^d Vol(\Omega)\lambda^{\frac{d}{2}}(1+o(1))$ Which in fact is a conjecture of *** in *** in published in 1910. This is a very amazing achievement of mathematicians, The realist meaning we can actually charge with the spectrum assessment. In…
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Sarnak conjecture, understand with standard model
Sarnak conjecture is a conjecture lies in the overlap of dynamic system and number theory. It is mainly focus on understanding the behavior of an entropy zero dynamic system by looking at the correlation of an observable and the Mobius function . We state it in a rigorous way: let $ (X,T)$ be an entropy zero topological dynamic system. Let the Mobius function be defined as $ \mu(n)=(-1)^t$, where $ $ is the number of different primes that occur in the decomposition of $ n$. Then for any continuo…
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Baragar-Bourgain-Gamburd-Sarnak conjecture
$latex M$ is the markov triple $(x,y,z)$: $x^2+y^2+z^2=xyz$ 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 solutions of (*) could be generated from $latex (1,1,1)$. I got a similar result for a similar algebraic equation 1 half years ago when considering a …
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Metric entropy 2
I am reading the article “ENTROPY THEORY OF GEODESIC FLOWS”. Now we focus on the upper semi-continuity of the metric entropy map. The object we investigate is $(X,T,\mu)$, where $\mu$ is a $T$-invariant measure. The insight that makes us interested in this kind of problem is a part of a variational problem, something about the existence of a certain object that combines a certain moduli space to make some quantity attain a critical value (maximum or minimum). The simplest example may…