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

${\displaystyle \left|q\alpha -p\right|<{\frac {1}{N+1}}}$

This theorem is a consequence of the pigeonhole principle, and the one who popularized its use ^{[citation needed]}.

An immediate consequence is that for a given irrational α, the equation

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}}$

is satisfied by infinitely many integers p and q. This corollary shows that the Thue–Siegel–Roth theorem provides the tightest possible bound, in the sense that approximation cannot be improved by increasing the exponent beyond 2.

References

• Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
• Wolfgang M. Schmidt.Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000

External links

• Dirichlet's Approximation Theorem at PlanetMath.