Distribution modulo one

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)

References

[a1]	E. Hlawka, "Theorie der Gleichverteilung" , B.I. Wissenschaftverlag Mannheim (1979)
[a2]	L. Kuipers, H. Niederreiter, "Uniform distribution of sequences" , Wiley (1974)