Dirichlet's approximation theorem

In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real number α and any positive integer N, there exists integers p and q such that 1 ≤ q ≤ N and

This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality

is satisfied by infinitely many integers p and q. This corollary also shows that the Thue–Siegel–Roth theorem, a result in the other direction, provides essentially the tightest possible bound, in the sense that the bound on rational approximation of algebraic numbers cannot be improved by increasing the exponent beyond 2.

The simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},...,\alpha _{d}}$ and a natural number ${\displaystyle N}$ then there are integers ${\displaystyle p_{1},...,p_{d},q\in \mathbb {Z} ,1\leq q\leq N}$ such that ${\displaystyle \left|\alpha _{i}-{\frac {p_{i}}{q}}\right|\leq {\frac {1}{qN^{1/d}}}.}$

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