Hi,
I´m looking for Chebyshev´s theorem which says that the inequality $x(k)y<3/k$ has infinitely many solutions, where $x(k)=x_0+k\alpha \pmod 1$, $\alpha$ is an irrational number, and $x_0,y\in S^1$. Does anybody know the exact formulation?
Hi, I´m looking for Chebyshev´s theorem which says that the inequality $x(k)y<3/k$ has infinitely many solutions, where $x(k)=x_0+k\alpha \pmod 1$, $\alpha$ is an irrational number, and $x_0,y\in S^1$. Does anybody know the exact formulation? 


This is a quantitative version of the onedimensional Kronecker's approximation theorem; see Hardy and Wright's classical text "An introduction to the Theory of Numbers", Theorem 440. You can also check this Wikipedia article. 


This looks very similar to the Equidistribution Theorem by Weyl 

