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The distribution of the fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335101.png" /> of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335102.png" /> of elements of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335103.png" />-dimensional Euclidean space in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335104.png" />-dimensional unit cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335105.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335106.png" /> denotes the fractional part of a number.
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The distribution of the fractional parts $\{P_j\} = (\{x_1^{(j)}\},\ldots,\{x_n^{(j)}\})$ of a sequence $P_j = (x_1^{(j)},\ldots,x_n^{(j)})$ of elements of the $n$-dimensional Euclidean space in the $n$-dimensional unit cube $E = \{ (x_1,\ldots,x_n) : 0 \le x_i < 1\,\ i=1,\ldots n\}$. Here $\{x\}$ denotes the [[fractional part of a number]] $x$.
  
The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335108.png" /> is said to be uniformly distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d0335109.png" /> if the equality
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The sequence $\{P_j\}$, $j=1,2,\ldots$ is said to be uniformly distributed in $E$ if the equality
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$$
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\lim_{m\rightarrow\infty} \frac{\phi_m(V)}{m} = |V|
 +
$$
 +
holds for any rectangle $V$, where $\phi_m(V)$ is the number of those points among the first $m$ members of the sequence which belong to $V$ and $|V|$ is the measure of $V$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351010.png" /></td> </tr></table>
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A sequence $P_j$, $j=1,2,\ldots$ is said to be uniformly distributed modulo one if the corresponding sequence of fractional parts $\{P_j\}$ is uniformly distributed in $E$.
 
 
holds for any rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351012.png" /> is the number of those points among the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351013.png" /> members of the sequence which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351015.png" /> is the measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351016.png" />.
 
 
 
A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351018.png" /> is said to be uniformly distributed modulo one if the corresponding sequence of fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351019.png" /> is uniformly distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351020.png" />.
 
  
 
===Weyl's criterion for higher-dimensional distribution modulo one.===
 
===Weyl's criterion for higher-dimensional distribution modulo one.===
A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351022.png" /> is uniformly distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351023.png" /> if and only if
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A sequence $P_j$, $j=1,2,\ldots$ is uniformly distributed in $E$ if and only if
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351024.png" /></td> </tr></table>
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\lim_{N\rightarrow\infty} \frac{1}{N} \sum_{j=1}^N \exp\left({ 2\pi i (a_1x_1^{(j)}+\cdots+a_n x_n^{(j)}) }\right) = 0
 
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$$
for any set of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351025.png" />. A particular case of this theorem is the [[Weyl criterion|Weyl criterion]] for a sequence of real numbers to be uniformly distributed modulo one. Weyl's criterion implies the following theorem of Kronecker: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351026.png" /> be real numbers that are linearly independent over the field of rational numbers, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351027.png" /> be arbitrary real numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351029.png" /> be positive numbers; then there are integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351031.png" /> such that
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for any set of integers $(a_1,\ldots,a_n) \ne (0,\ldots,0)$. A particular case of this theorem is the [[Weyl criterion]] for a sequence of real numbers to be uniformly distributed modulo one. Weyl's criterion implies the following theorem of Kronecker: Let $\theta_1,\ldots,\theta_n,1$ be real numbers that are linearly independent over the field of rational numbers, let $\alpha_1,\ldots,\alpha_n$ be arbitrary real numbers and let $N$ and $\epsilon$ be positive numbers; then there are integers $m$ and $p_1,\ldots,p_n$ such that
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351032.png" /></td> </tr></table>
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m>N\,,\ \ \ |m\theta_i-p_i-\alpha_i| < \epsilon
 
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$$
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351033.png" />. In other words, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033510/d03351035.png" /> is uniformly distributed modulo one.
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for all $i=1,\ldots,n$. Indeed, the sequence $m\theta = (m\theta_1,\ldots,m\theta_n)$, $m=1,2,\ldots$ is uniformly distributed modulo one.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels,  "An introduction to diophantine approximation" , Cambridge Univ. Press  (1957)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels,  "An introduction to diophantine approximation" , Cambridge Univ. Press  (1957)</TD></TR>
 +
</table>
  
 +
====Comments====
 +
For additional references see [[Distribution modulo one]].
  
====Comments====
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{{TEX|done}}
For additional references see [[Distribution modulo one|Distribution modulo one]].
 

Latest revision as of 12:10, 13 March 2018

The distribution of the fractional parts $\{P_j\} = (\{x_1^{(j)}\},\ldots,\{x_n^{(j)}\})$ of a sequence $P_j = (x_1^{(j)},\ldots,x_n^{(j)})$ of elements of the $n$-dimensional Euclidean space in the $n$-dimensional unit cube $E = \{ (x_1,\ldots,x_n) : 0 \le x_i < 1\,\ i=1,\ldots n\}$. Here $\{x\}$ denotes the fractional part of a number $x$.

The sequence $\{P_j\}$, $j=1,2,\ldots$ is said to be uniformly distributed in $E$ if the equality $$ \lim_{m\rightarrow\infty} \frac{\phi_m(V)}{m} = |V| $$ holds for any rectangle $V$, where $\phi_m(V)$ is the number of those points among the first $m$ members of the sequence which belong to $V$ and $|V|$ is the measure of $V$.

A sequence $P_j$, $j=1,2,\ldots$ is said to be uniformly distributed modulo one if the corresponding sequence of fractional parts $\{P_j\}$ is uniformly distributed in $E$.

Weyl's criterion for higher-dimensional distribution modulo one.

A sequence $P_j$, $j=1,2,\ldots$ is uniformly distributed in $E$ if and only if $$ \lim_{N\rightarrow\infty} \frac{1}{N} \sum_{j=1}^N \exp\left({ 2\pi i (a_1x_1^{(j)}+\cdots+a_n x_n^{(j)}) }\right) = 0 $$ for any set of integers $(a_1,\ldots,a_n) \ne (0,\ldots,0)$. A particular case of this theorem is the Weyl criterion for a sequence of real numbers to be uniformly distributed modulo one. Weyl's criterion implies the following theorem of Kronecker: Let $\theta_1,\ldots,\theta_n,1$ be real numbers that are linearly independent over the field of rational numbers, let $\alpha_1,\ldots,\alpha_n$ be arbitrary real numbers and let $N$ and $\epsilon$ be positive numbers; then there are integers $m$ and $p_1,\ldots,p_n$ such that $$ m>N\,,\ \ \ |m\theta_i-p_i-\alpha_i| < \epsilon $$ for all $i=1,\ldots,n$. Indeed, the sequence $m\theta = (m\theta_1,\ldots,m\theta_n)$, $m=1,2,\ldots$ is uniformly distributed modulo one.

References

[1] J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957)

Comments

For additional references see Distribution modulo one.

How to Cite This Entry:
Distribution modulo one, higher-dimensional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distribution_modulo_one,_higher-dimensional&oldid=42944
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article