The theory of uniform distribution modulo 1 deals with the distribution behavior of sequences of real numbers. A sequence is uniformly distributed modulo 1, if the relative number of sequence elements converge at an interval to the length of this interval.
Let be a sequence of real numbers. For numbers denote the number of those followers with index less than or equal to, the fraction is in the interval. In mathematical notation:
Under the fraction of a number is taken to mean the number itself minus the nearest whole number ( for example, the fraction, and the fractional part ). The fractional part of a number is always in the interval.
The sequence is now uniformly distributed modulo 1, if the relative number of sequence elements in the interval targeted for each interval to the length of the interval. In mathematical notation: ie uniformly distributed modulo 1 if and only if
Intuitively, this means that the sequence is uniformly distributed in the interval ( hence the term " uniformly distributed modulo 1").
An important criterion to check whether a sequence is uniformly distributed modulo 1 or not, is the Weyl criterion, first proved by Hermann Weyl in 1916. A sequence is uniformly distributed modulo 1 if and only if
The proof is based on that the indicator functions appearing in the definition of uniform distribution modulo 1, according to the approximation theorem of Weierstrass can be approximated accurately by continuous functions by trigonometric polynomials and these arbitrary.
The following consequences are uniformly distributed modulo 1:
- If and only if is irrational.
- Wherein a non- constant polynomial denoted which has at least an irrational coefficients.
- If and only if a normal number in base 2.
Since the sequence of irrational uniformly distributed modulo 1, must be asymptotically about elements of the sequence in each interval by definition. In particular, therefore each interval must contain infinitely many elements of the sequence: the sequence is therefore dense in the interval. This is the so-called approximation theorem of Kronecker, creating a connection between uniform distribution modulo 1 and diophantine approximation is indicated (see Dirichlet's approximation theorem ).