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 .

definition

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: ${\displaystyle x_{1},x_{2},\dots }$ ${\displaystyle a,b}$ ${\displaystyle 0\leq a<b\leq 1}$ ${\displaystyle A([a,b),N)}$ ${\displaystyle N}$ ${\displaystyle [a,b)}$

${\displaystyle A([a,b),N):=\#\left\{1\leq n\leq N:~\{x_{n}\}\in [a,b)\right\}}$.

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 . ${\displaystyle \{x\}}$ ${\displaystyle x}$ ${\displaystyle \{1{,}4142\}=1{,}4142-1=0{,}4142}$ ${\displaystyle \{-0{,}7\}=-0{,}7-(-1)=0{,}3}$ ${\displaystyle [0,1)}$

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 ${\displaystyle x_{1},x_{2},\dots }$ ${\displaystyle [a,b)\subset [0,1)}$ ${\displaystyle x_{1},x_{2},\dots }$

${\displaystyle \lim _{N\to \infty }{\frac {A([a,b),N)}{N}}=b-a}$    applies to all numbers with . ${\displaystyle a,b}$ ${\displaystyle 0\leq a<b\leq 1}$

In clear terms, this means that the sequence is evenly distributed in the interval (hence the term "uniformly distributed modulo 1"). ${\displaystyle x_{1},x_{2},\dots }$ ${\displaystyle [0,1)}$

properties

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 ${\displaystyle x_{1},x_{2},\dots }$ ${\displaystyle x_{1},x_{2},\dots }$

${\displaystyle \lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}e^{2\pi ihx_{n}}=0}$     applies to all . ${\displaystyle h\in \mathbb {Z} \backslash \{0\}}$

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 .

Examples

The following consequences are evenly distributed modulo 1:

• ${\displaystyle (n\alpha )_{n\geq 1}}$    exactly when is irrational . ${\displaystyle \alpha }$

• ${\displaystyle (n^{\sigma }\log ^{\tau }n)_{n\geq 1}}$    For ${\displaystyle 0<\sigma <1,~\tau \in \mathbb {R} }$

• ${\displaystyle (p(n))_{n\geq 1}}$    where denotes a non-constant polynomial which has at least one irrational coefficient. ${\displaystyle p(x)}$

• ${\displaystyle (2^{n}\alpha )_{n\geq 1}}$   if and only if is a normal number to base 2. ${\displaystyle \alpha }$

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 ). ${\displaystyle (n\alpha )_{n\geq 1}}$ ${\displaystyle \alpha }$ ${\displaystyle [a,b)}$ ${\displaystyle N(b-a)}$ ${\displaystyle n\alpha }$ ${\displaystyle [0,1)}$

literature

• Edmund Hlawka: Theory of equal distribution . BI-Wissenschaftsverlag, 1979. ISBN 3-411-01565-9
• Lauwerens Kuipers and Harald Niederreiter: Uniform distribution of sequences . Dover Publications, 2002. ISBN 0-486-45019-8