Distribution modulo one

The distribution of the fractional parts $\{\alpha_j\}$ of a sequence of real numbers $\alpha_j$, $j=1,2,\dots,$ in the unit interval $[0,1)$. The sequence of fractional parts $\{\alpha_j\}$, $j=1,2,\dots,$ is called uniformly distributed in $[0,1)$ if the equality

$$\lim_{n\to\infty}\frac{\phi_n(a,b)}{n}=b-a$$

holds for any interval $[a,b)\subset[0,1)$, where $\phi_n(a,b)$ is the number of terms among the first $n$ members of $\{\alpha_j\}$, $j=1,2,\dots,$ which belong to $[a,b)$. In this case the sequence $\alpha_j$, $j=1,2,\dots,$ 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 $\{\alpha_j\}$, $j=1,2,\dots,$ is uniformly distributed in the unit interval $[0,1)$ if and only if

$$\lim_{n\to\infty}\frac1n\sum_{j=1}^nf(\{\alpha_j\})=\int\limits_0^1f(x)dx$$

for any function $f$ that is Riemann integrable on $[0,1]$. This is equivalent to the following. In order that a sequence $\alpha_j$, $j=1,2,\dots,$ be uniformly distributed modulo one, it is necessary and sufficient that

$$\lim_{n\to\infty}\frac1n\sum_{j=1}^ne^{2\pi im\alpha_j}=0$$

for any integer $m\neq0$. It follows from Weyl's criterion and his estimates for trigonometric sums involving a polynomial $f$,

$$\sum_{x=1}^pe^{2\pi if(x)},$$

that the sequence $\{f(n)\}$, $n=1,2,\dots,$ of fractional parts is uniformly distributed in $[0,1)$ provided that at least one coefficient $a_s$, $1\leq s\leq k$, of the polynomial

$$f(x)=a_kx^k+\dotsb+a_1x$$

is irrational.

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

$$D_n=\sup_{0\leq a<b\leq1}\left|\frac{\phi_n(a,b)}{n}-(b-a)\right|,$$

called the discrepancy of the first $n$ members of the sequence $\{\alpha_j\}$, $j=1,2,\dots$ (see [2], [3]).

References

[1]	H. Weyl, "Ueber die Gleichverteilung von Zahlen mod Eins" Math. Ann. , 77 (1916) pp. 313–352 Zbl 46.0278.06
[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)
[a1]	E. Hlawka, "Theorie der Gleichverteilung" , B.I. Wissenschaftverlag Mannheim (1979)
[a2]	L. Kuipers, H. Niederreiter, "Uniform distribution of sequences" , Wiley (1974) Zbl 0281.10001; repr. Dover (2006) ISBN 0-486-45019-8