Uniform distribution modulo 1
The theory of uniform distribution modulo 1 deals with the distribution behavior of sequences of real numbers . A sequence is called uniformly distributed modulo 1 if the relative number of sequence elements in an interval converges to the length of this interval .
Be a sequence of real numbers. For numbers with denote the number of those sequence members with an index less than or equal to whose fraction lies in the interval . In mathematical notation:
The fraction of a number is understood to mean the number itself minus the next lower whole number (for example, the fraction is , and the fraction ). The fraction of a number is always in the interval .
The sequence is now called uniformly distributed modulo 1 , if for each interval the relative number of sequence elements in this interval tends towards the length of the interval. In mathematical notation: means uniformly distributed modulo 1 if and only if
- applies to all numbers with .
In clear terms, this means that the sequence is evenly distributed in the interval (hence the term "uniformly distributed modulo 1").
An important criterion for checking whether a sequence is uniformly distributed modulo 1 or not is Weyl's criterion , first proven by Hermann Weyl in 1916. A sequence is uniformly distributed modulo 1 if and only if
- applies to all .
The proof is based on the fact that the indicator functions appearing in the definition of the uniform distribution modulo 1 by continuous functions , and that these can be approximated as precisely as desired by trigonometric polynomials according to Weierstrass's approximation theorem .
The following consequences are evenly distributed modulo 1:
- exactly when is irrational .
- where denotes a non-constant polynomial which has at least one irrational coefficient.
- if and only if is a normal number to base 2.
Since the sequence for irrational things is uniformly distributed modulo 1, according to the definition there must be asymptotically about elements of the sequence in every interval . In particular, every interval must therefore contain infinitely many elements of the sequence: the sequence is therefore dense in the interval . This is Kronecker's so-called approximation theorem , which indicates a connection between uniform distribution modulo 1 and Diophantine approximation (see Dirichlet's approximation theorem ).