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Kronecker's Approximation Theorem

Kronecker's approximation theorem states that if theta is a given irrational number, then the sequence of numbers {ntheta}, where {x}=x-|_x_|, is dense in the unit interval. Explicitly, given any alpha, 0<=alpha<=1, and given any epsilon>0, there exists a positive integer k such that

|{ktheta}-alpha|<epsilon.

Therefore, if h=|_ktheta_|, it follows that |ktheta-h-alpha|<epsilon. The restriction on alpha can be removed as follows. Given any real alpha, any irrational theta, and any epsilon>0, there exist integers h and k with k>0 such that

|ktheta-h-alpha|<epsilon.

See also

Rational Approximation

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References

Apostol, T. M. "Kronecker's Approximation Theorem: The One-Dimensional Case" and "Extension of Kronecker's Theorem to Simultaneous Approximation." §7.4 and 7.5 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 148-155, 1997.Montgomery, H. L. "Harmonic Analysis as Found in Analytic Number Theory." In Twentieth Century Harmonic Analysis--A Celebration. Proceedings of the NATO Advanced Study Institute Held in Il Ciocco, July 2-15, 2000 (Ed. J. S. Byrnes). Dordrecht, Netherlands: Kluwer, pp. 271-293, 2001.

Referenced on Wolfram|Alpha

Kronecker's Approximation Theorem

Cite this as:

Weisstein, Eric W. "Kronecker's Approximation Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KroneckersApproximationTheorem.html

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