Weyl's Criterion

A sequence {x_1,x_2,...} is equidistributed iff


for each m=1, 2, .... A consequence of this result is that the sequence {frac(nx)} is equidistributed, and hence dense, in the interval [0,1] for irrational x, where n=1, 2, ... and frac(x) is the fractional part of x (Finch 2003).

See also

Equidistributed Sequence, Erdős-Turán Discrepancy Bound, Fractional Part, Ramanujan's Sum, Weyl Sum

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Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.Cassels, J. W. S. An Introduction to Diophantine Analysis. Cambridge, England: Cambridge University Press, 1965.Finch, S. R. "Powers of 3/2 Modulo One." §2.30.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 194-199, 2003.Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, pp. 7 and 226, 1974.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.Pólya, G. and Szegö, G. Problems and Theorems in Analysis I. New York: Springer-Verlag, 1972.Radin, C. Miles of Tiles. Providence, RI: Amer. Math. Soc., pp. 79-80, 1999.Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 155-156 and 254, 1991.Weyl, H. "Über ein Problem aus dem Gebiete der diophantischen Approximationen." Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 234-244, 1914. Reprinted in Gesammelte Abhandlungen, Band I. Berlin: Springer-Verlag, pp. 487-497, 1968.Weyl, H. "Über die Gleichverteilung von Zahlen mod. Eins." Math. Ann. 77, 313-352, 1916. Reprinted in Gesammelte Abhandlungen, Band I. Berlin: Springer-Verlag, pp. 563-599, 1968. Also reprinted in Selecta Hermann Weyl. Basel, Switzerland: Birkhäuser, pp. 111-147, 1956.

Referenced on Wolfram|Alpha

Weyl's Criterion

Cite this as:

Weisstein, Eric W. "Weyl's Criterion." From MathWorld--A Wolfram Web Resource.

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