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

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 ${\ displaystyle x_ {1}, x_ {2}, \ dots}$ ${\ displaystyle [a, b) \ subset [0,1)}$ ${\ displaystyle x_ {1}, x_ {2}, \ dots}$ ${\ displaystyle \ lim _ {N \ to \ infty} {\ frac {A ([a, b), N)} {N}} = ba}$ applies to all numbers with .${\ displaystyle a, b}$ ${\ displaystyle 0 \ leq a 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 ). ${\ displaystyle (n \ alpha) _ {n \ geq 1}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle [a, b)}$ ${\ displaystyle N (ba)}$ ${\ displaystyle n \ alpha}$ ${\ displaystyle [0,1)}$ ## 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