Namespaces
Variants
Actions

Distribution modulo one

From Encyclopedia of Mathematics
Revision as of 17:05, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

The distribution of the fractional parts of a sequence of real numbers , in the unit interval . The sequence of fractional parts , is called uniformly distributed in if the equality

holds for any interval , where is the number of terms among the first members of , which belong to . In this case the sequence , is said to be uniformly distributed modulo one.

Weyl's criterion (see [1]) for a distribution modulo one to be uniform: An infinite sequence of fractional parts , is uniformly distributed in the unit interval if and only if

for any function that is Riemann integrable on . This is equivalent to the following. In order that a sequence , be uniformly distributed modulo one, it is necessary and sufficient that

for any integer . It follows from Weyl's criterion and his estimates for trigonometric sums involving a polynomial ,

that the sequence , of fractional parts is uniformly distributed in provided that at least one coefficient , , of the polynomial

is irrational.

The concept of uniform distribution modulo one can be made quantitative by means of the quantity

called the discrepancy of the first members of the sequence , (see [2], [3]).

References

[1] H. Weyl, "Ueber die Gleichverteilung von Zahlen mod Eins" Math. Ann. , 77 (1916) pp. 313–352
[2] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
[3] L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1)


Comments

References

[a1] E. Hlawka, "Theorie der Gleichverteilung" , B.I. Wissenschaftverlag Mannheim (1979)
[a2] L. Kuipers, H. Niederreiter, "Uniform distribution of sequences" , Wiley (1974)
How to Cite This Entry:
Distribution modulo one. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distribution_modulo_one&oldid=33478
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article