| Version current |
Version 2 |
| Let $\{u_n\}$ be a sequence of real numbers. For |
Let $\{u_n\}$ be a sequence of real numbers. For |
| $0\leq\alpha<\beta\leq 1$ put |
$0\leq\alpha<\beta\leq 1$ put |
| \begin{equation*} |
\begin{equation*} |
| Z(N,\alpha,\beta)=\operatorname{card}\{n\in[1..N] : \alpha \leq |
Z(N,\alpha,\beta)=\operatorname{card}\{n\in[1..N] : \alpha \leq |
| (u_n \bmod 1)< \beta \}. |
(u_n \bmod 1)< \beta \}. |
| \end{equation*} |
\end{equation*} |
| The sequence $\{u_n\}$ is \emph{uniformly distributed modulo $1$} |
The sequence $\{u_n\}$ is \emph{uniformly distributed modulo $1$} |
| if |
|
| \begin{equation*}\label{eq:modcond} |
\begin{equation*}\label{eq:modcond} |
| \lim_{N\to\infty} \frac{1}{N} Z(N,\alpha,\beta)=\beta-\alpha |
\lim_{N\to\infty} \frac{1}{N} Z(N,\alpha,\beta)=\beta-\alpha |
| \end{equation*} |
\end{equation*} |
| for all $0\leq\alpha<\beta\leq 1$. In other words a sequence is |
for all $0\leq\alpha<\beta\leq 1$. In other words a sequence is |
| uniformly distributed modulo $1$ if each subinterval of $[0,1]$ |
uniformly distributed modulo $1$ if each subinterval of $[0,1]$ |
| gets its ``fair share'' of fractional parts of $\{u_n\}$. |
gets its ``fair share'' of fractional parts of $\{u_n\}$. |
|
|
| More generally, a sequence $\{u_n\}$ of points in a finite measure |
More generally, a sequence $\{u_n\}$ of points in a finite measure |
| space $(X,\mathcal{A},\mu)$ is uniformly distributed with respect |
space $(X,\mathcal{A},\mu)$ is uniformly distributed with respect |
| to a family of sets $\mathcal{F}\subset\mathcal{A}$ if |
to a family of sets $\mathcal{F}\subset\mathcal{A}$ if |
| \begin{equation*} |
\begin{equation*} |
|
\lim_{N\to\infty} \frac{\operatorname{card}\{n\in[1..N] :u_n\in
|
\lim_{N\to\infty} \frac{\operatorname{card}\{1\leq n\leq N :u_n\in
|
| S\}}{N}=\frac{\mu(S)}{\mu(X)}\qquad\text{for every |
S\}}{N}=\frac{\mu(S)}{\mu(X)}\qquad\text{for every |
| }S\in\mathcal{F}. |
}S\in\mathcal{F}. |
| \end{equation*} |
\end{equation*} |
|
|
| \begin{thebibliography}{1} |
\begin{thebibliography}{1} |
|
|
| \bibitem{cite:chen_irreg_dist} |
\bibitem{cite:chen_irreg_dist} |
| William Chen. |
William Chen. |
| \newblock Lectures on irregularities of point distribution. |
\newblock Lectures on irregularities of point distribution. |
| \newblock Available at \PMlinkexternal{http://www.maths.mq.edu.au/~wchen/ln.html}{http://www.maths.mq.edu.au/~wchen/ln.html}, 2000. |
\newblock Available at \PMlinkexternal{http://www.maths.mq.edu.au/~wchen/ln.html}{http://www.maths.mq.edu.au/~wchen/ln.html}, 2000. |
|
|
| \bibitem{cite:montgomery_tenlect} |
\bibitem{cite:montgomery_tenlect} |
| Hugh~L. Montgomery. |
Hugh~L. Montgomery. |
| \newblock {\em Ten Lectures on the Interface Between Analytic Number Theory and |
\newblock {\em Ten Lectures on the Interface Between Analytic Number Theory and |
| Harmonic Analysis}, volume~84 of {\em Regional Conference Series in |
Harmonic Analysis}, volume~84 of {\em Regional Conference Series in |
| Mathematics}. |
Mathematics}. |
| \newblock AMS, 1994. |
\newblock AMS, 1994. |
| \newblock \PMlinkexternal{Zbl 0814.11001}{http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0814.11001}. |
\newblock \PMlinkexternal{Zbl 0814.11001}{http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0814.11001}. |
|
|
| \end{thebibliography} |
\end{thebibliography} |
