Diophantus approximation

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…

An important theorem in Diophantine approximation is the theorem of Liuoville: **Liuoville Theorem** If x is a algebraic number of degree $latex n$ over the rational number then there exists a constant $latex c(x) > 0$ such that:$latex \left|x-{\frac {p}{q}}\right|>{\frac {c(x)}{q^{{n}}}}$ holds for every integer $latex p,q\in N^*$ where $latex q>0$. This theorem explain a phenomenon, the approximation of algebraic number by rational number could not be very well. Which was generated la…

this note will talk about the Ostrowski representation and approximation by continue fraction. As well-known,by the Weyl criterion,$latex \{n\alpha\}$ is uniformly distribution in $latex [0,1]$ iff $latex \alpha\in R-Q$. i.e. we have:$latex \forall 0\leq a\leq b\leq 1$,we have: $latex \lim_{N\to \infty}|\{1\leq n\leq N|\{n\alpha\}\in [a,b]\}|=(b-a)N+o(N)$. but this will not give the effective version.i.e. we do not the the more information about the decay of $latex o(N)$. we will give a approach…