Therefore, if , it follows that . The restriction on can be removed as follows. Given any real , any irrational , and any , there exist integers and with such that
Kronecker's Approximation Theorem
See alsoRational Approximation
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ReferencesApostol, 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|AlphaKronecker's Approximation Theorem
Cite this as:
Weisstein, Eric W. "Kronecker's Approximation Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KroneckersApproximationTheorem.html