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The distribution of the fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d0335001.png" /> of a sequence of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d0335002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d0335003.png" /> in the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d0335004.png" />. The sequence of fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d0335005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d0335006.png" /> is called uniformly distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d0335007.png" /> if the equality
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The distribution of the fractional parts $\{\alpha_j\}$ of a sequence of real numbers $\alpha_j$, $j=1,2,\dots,$ in the unit interval $[0,1)$. The sequence of fractional parts $\{\alpha_j\}$, $j=1,2,\dots,$ is called uniformly distributed in $[0,1)$ if the equality
  
<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/d033500/d0335008.png" /></td> </tr></table>
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$$\lim_{n\to\infty}\frac{\phi_n(a,b)}{n}=b-a$$
  
holds for any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d0335009.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350010.png" /> is the number of terms among the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350011.png" /> members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350013.png" /> which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350014.png" />. In this case the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350016.png" /> is said to be uniformly distributed modulo one.
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holds for any interval $[a,b)\subset[0,1)$, where $\phi_n(a,b)$ is the number of terms among the first $n$ members of $\{\alpha_j\}$, $j=1,2,\dots,$ which belong to $[a,b)$. In this case the sequence $\alpha_j$, $j=1,2,\dots,$ is said to be uniformly distributed modulo one.
  
Weyl's criterion (see [[#References|[1]]]) for a distribution modulo one to be uniform: An infinite sequence of fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350018.png" /> is uniformly distributed in the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350019.png" /> if and only if
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[[Weyl criterion|Weyl's criterion]] (see [[#References|[1]]]) for a distribution modulo one to be uniform: An infinite sequence of fractional parts $\{\alpha_j\}$, $j=1,2,\dots,$ is uniformly distributed in the unit interval $[0,1)$ if and only if
  
<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/d033500/d03350020.png" /></td> </tr></table>
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$$\lim_{n\to\infty}\frac1n\sum_{j=1}^nf(\{\alpha_j\})=\int\limits_0^1f(x)dx$$
  
for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350021.png" /> that is Riemann integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350022.png" />. This is equivalent to the following. In order that a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350024.png" /> be uniformly distributed modulo one, it is necessary and sufficient that
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for any function $f$ that is Riemann integrable on $[0,1]$. This is equivalent to the following. In order that a sequence $\alpha_j$, $j=1,2,\dots,$ be uniformly distributed modulo one, it is necessary and sufficient that
  
<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/d033500/d03350025.png" /></td> </tr></table>
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$$\lim_{n\to\infty}\frac1n\sum_{j=1}^ne^{2\pi im\alpha_j}=0$$
  
for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350026.png" />. It follows from Weyl's criterion and his estimates for trigonometric sums involving a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350027.png" />,
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for any integer $m\neq0$. It follows from Weyl's criterion and his estimates for trigonometric sums involving a polynomial $f$,
  
<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/d033500/d03350028.png" /></td> </tr></table>
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$$\sum_{x=1}^pe^{2\pi if(x)},$$
  
that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350030.png" /> of fractional parts is uniformly distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350031.png" /> provided that at least one coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350033.png" />, of the polynomial
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that the sequence $\{f(n)\}$, $n=1,2,\dots,$ of fractional parts is uniformly distributed in $[0,1)$ provided that at least one coefficient $a_s$, $1\leq s\leq k$, of the polynomial
  
<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/d033500/d03350034.png" /></td> </tr></table>
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$$f(x)=a_kx^k+\dotsb+a_1x$$
  
 
is irrational.
 
is irrational.
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The concept of uniform distribution modulo one can be made quantitative by means of the quantity
 
The concept of uniform distribution modulo one can be made quantitative by means of the quantity
  
<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/d033500/d03350035.png" /></td> </tr></table>
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$$D_n=\sup_{0\leq a<b\leq1}\left|\frac{\phi_n(a,b)}{n}-(b-a)\right|,$$
  
called the [[Discrepancy|discrepancy]] of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350036.png" /> members of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033500/d03350038.png" /> (see [[#References|[2]]], [[#References|[3]]]).
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called the [[Discrepancy|discrepancy]] of the first $n$ members of the sequence $\{\alpha_j\}$, $j=1,2,\dots$ (see [[#References|[2]]], [[#References|[3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl,   "Ueber die Gleichverteilung von Zahlen mod Eins"  ''Math. Ann.'' , '''77'''  (1916)  pp. 313–352</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Vinogradov,   "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.-K. Hua,   "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , ''Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen'' , '''1''' :  2  (1959)  (Heft 13, Teil 1)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl, "Ueber die Gleichverteilung von Zahlen mod Eins"  ''Math. Ann.'' , '''77'''  (1916)  pp. 313–352 {{ZBL|46.0278.06}}</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR>
 
+
<TR><TD valign="top">[3]</TD> <TD valign="top"> L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , ''Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen'' , '''1''' :  2  (1959)  (Heft 13, Teil 1)</TD></TR>
====Comments====
+
<TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Hlawka, "Theorie der Gleichverteilung" , B.I. Wissenschaftverlag Mannheim  (1979)</TD></TR>
 
+
<TR><TD valign="top">[a2]</TD> <TD valign="top"> L. Kuipers, H. Niederreiter, "Uniform distribution of sequences" , Wiley  (1974) {{ZBL|0281.10001}}; repr. Dover (2006) {{ISBN|0-486-45019-8}}
 
+
</TD></TR>
====References====
+
</table>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Hlawka,   "Theorie der Gleichverteilung" , B.I. Wissenschaftverlag Mannheim  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L. Kuipers,   H. Niederreiter,   "Uniform distribution of sequences" , Wiley  (1974)</TD></TR></table>
 

Latest revision as of 19:28, 18 May 2024

The distribution of the fractional parts $\{\alpha_j\}$ of a sequence of real numbers $\alpha_j$, $j=1,2,\dots,$ in the unit interval $[0,1)$. The sequence of fractional parts $\{\alpha_j\}$, $j=1,2,\dots,$ is called uniformly distributed in $[0,1)$ if the equality

$$\lim_{n\to\infty}\frac{\phi_n(a,b)}{n}=b-a$$

holds for any interval $[a,b)\subset[0,1)$, where $\phi_n(a,b)$ is the number of terms among the first $n$ members of $\{\alpha_j\}$, $j=1,2,\dots,$ which belong to $[a,b)$. In this case the sequence $\alpha_j$, $j=1,2,\dots,$ is said to be uniformly distributed modulo one.

Weyl's criterion (see [1]) for a distribution modulo one to be uniform: An infinite sequence of fractional parts $\{\alpha_j\}$, $j=1,2,\dots,$ is uniformly distributed in the unit interval $[0,1)$ if and only if

$$\lim_{n\to\infty}\frac1n\sum_{j=1}^nf(\{\alpha_j\})=\int\limits_0^1f(x)dx$$

for any function $f$ that is Riemann integrable on $[0,1]$. This is equivalent to the following. In order that a sequence $\alpha_j$, $j=1,2,\dots,$ be uniformly distributed modulo one, it is necessary and sufficient that

$$\lim_{n\to\infty}\frac1n\sum_{j=1}^ne^{2\pi im\alpha_j}=0$$

for any integer $m\neq0$. It follows from Weyl's criterion and his estimates for trigonometric sums involving a polynomial $f$,

$$\sum_{x=1}^pe^{2\pi if(x)},$$

that the sequence $\{f(n)\}$, $n=1,2,\dots,$ of fractional parts is uniformly distributed in $[0,1)$ provided that at least one coefficient $a_s$, $1\leq s\leq k$, of the polynomial

$$f(x)=a_kx^k+\dotsb+a_1x$$

is irrational.

The concept of uniform distribution modulo one can be made quantitative by means of the quantity

$$D_n=\sup_{0\leq a<b\leq1}\left|\frac{\phi_n(a,b)}{n}-(b-a)\right|,$$

called the discrepancy of the first $n$ members of the sequence $\{\alpha_j\}$, $j=1,2,\dots$ (see [2], [3]).

References

[1] H. Weyl, "Ueber die Gleichverteilung von Zahlen mod Eins" Math. Ann. , 77 (1916) pp. 313–352 Zbl 46.0278.06
[2] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
[3] L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1)
[a1] E. Hlawka, "Theorie der Gleichverteilung" , B.I. Wissenschaftverlag Mannheim (1979)
[a2] L. Kuipers, H. Niederreiter, "Uniform distribution of sequences" , Wiley (1974) Zbl 0281.10001; repr. Dover (2006) ISBN 0-486-45019-8
How to Cite This Entry:
Distribution modulo one. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distribution_modulo_one&oldid=13739
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article