View source for Weyl criterion

A fundamental criterion used to solve the problem of the [[Uniform distribution|uniform distribution]] of an infinite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977001.png" /> of arbitrary real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977002.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977003.png" />, i.e. to establish the existence of the limit

<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/w/w097/w097700/w0977004.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977006.png" /> is the fractional part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977007.png" /> (cf. [[Fractional part of a number|Fractional part of a number]]). According to Weyl's criterion, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977008.png" /> is uniformly distributed modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977009.png" /> 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/w/w097/w097700/w09770010.png" /></td> </tr></table>

for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w09770011.png" />. Demonstrated in 1916 by H. Weyl. See [[Weyl method|Weyl method]].

====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>