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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?

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This is a quantitative version of the one-dimensional 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.

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This looks very similar to the Equidistribution Theorem by Weyl

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