Given any real number 
and any positive integer 
,
there exist integers 
and 
with 
such that
 |
A slightly weaker form of the theorem states that for every real 
, there exist integers 
and 
with 
and 
such that
 |
Dirichlet's Approximation Theorem
See also
Hurwitz's Irrational Number Theorem, Irrationality Measure, Liouville's Approximation Theorem, Rational Approximation, Roth's TheoremExplore with Wolfram|Alpha
References
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 TheoremCite this as:
Weisstein, Eric W. "Dirichlet's Approximation Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletsApproximationTheorem.html