Dirichlet's Approximation Theorem

Given any real number theta and any positive integer N, there exist integers h and k with 0<k<=N such that


A slightly weaker form of the theorem states that for every real theta, there exist integers h and k with k>0 and (h,k)=1 such that


See also

Hurwitz's Irrational Number Theorem, Irrationality Measure, Liouville's Approximation Theorem, Rational Approximation, Roth's Theorem

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Apostol, T. M. "Dirichlet's Approximation Theorem." §7.2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 143-145, 1997.

Referenced on Wolfram|Alpha

Dirichlet's Approximation Theorem

Cite this as:

Weisstein, Eric W. "Dirichlet's Approximation Theorem." From MathWorld--A Wolfram Web Resource.

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