| Version current |
Version 3 |
| Let $\{u_n\}$ be a sequence of real numbers. Then $\{u_n\}$ is |
Let $\{u_n\}$ be a sequence of real numbers. Then $\{u_n\}$ is |
| uniformly distributed modulo $1$ if and only if |
uniformly distributed modulo $1$ if and only if |
| \begin{equation*} |
\begin{equation*} |
| \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N e(k u_n)=0 |
\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N e(k u_n)=0 |
| \end{equation*} |
\end{equation*} |
| for every nonzero integer $k$, where $e(x)=\exp(2\pi i x)$. |
for every nonzero integer $k$, where $e(x)=\exp(2\pi i x)$. |
|
|
| Weyl's criterion reduces the problem of uniform distribution of |
Weyl's criterion reduces the problem of uniform distribution of |
| sequences to the problem of estimating certain exponential sums. |
sequences to the problem of estimating certain exponential sums. |
| Whereas the problem of estimating a family of exponential sums |
Whereas the problem of estimating a family of exponential sums |
| might seem harder at first, the exponential map has the |
might seem harder at first, the exponential map has the |
| multiplicative property which often makes the problem easier. |
multiplicative property which often makes the problem easier. |
|
|
| \emph{Example:} If $x$ is irrational, then the sequence $\{nx\}$ |
\emph{Example:} If $x$ is irrational, then the sequence $\{nx\}$ |
| is uniformly distributed modulo $1$. Proof: |
is uniformly distributed modulo $1$. Proof: |
| \begin{equation*} |
\begin{equation*} |
| \abs{\sum_{n=1}^{N} e(k n |
\abs{\sum_{n=1}^{N} e(k n |
| x)}=\abs{\frac{e(k(N+1)x)-e(kx)}{e(kx)-1}}\leq |
x)}=\abs{\frac{e(k(N+1)x)-e(kx)}{e(kx)-1}}\leq |
| \frac{2}{\abs{\,e(kx)-1}}=O_k(1) |
\frac{2}{\abs{\,e(kx)-1}}=O_k(1) |
| \end{equation*} |
\end{equation*} |
| because the irrationality of $x$ implies $e(kx)\neq 1$. |
because the irrationality of $x$ implies $e(kx)\neq 1$. |
|
|
| \begin{thebibliography}{1} |
\begin{thebibliography}{1} |
|
|
| \bibitem{cite:karatsuba_ant} |
\bibitem{cite:karatsuba_ant} |
| Ð?.~Ð?. ÐаÑаÑÑба. |
\newblock {\em |
|
\newblock {\em ÐÑ?Ð½Ð¾Ð²Ñ Ð°Ð½Ð°Ð»Ð¸ÑиÑеÑ?кой ÑеоÑии ÑиÑ?ел}.
|
\newblock |
|
\newblock Ð?аÑка, 1983.
|
, 1983. |
| \newblock \PMlinkexternal{Zbl 0428.10019}{http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0428.10019}. |
\newblock \PMlinkexternal{Zbl 0428.10019}{http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0428.10019}. |
| \newblock For English translation see \cite{cite:karatsuba_ant_eng}. |
|
|
|
| \bibitem{cite:karatsuba_ant_eng} |
\bibitem{cite:karatsuba_ant_eng} |
| A.~A. Karatsuba. |
A.~A. Karatsuba. |
| \newblock {\em Basic analytic number theory}. |
\newblock {\em Basic analytic number theory}. |
| \newblock Springer-Verlag, 1993. |
\newblock Springer-Verlag, 1993. |
| \newblock \PMlinkexternal{Zbl 0767.11001}{http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0767.11001}. |
\newblock \PMlinkexternal{Zbl 0767.11001}{http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0767.11001}. |
| %\newblock This is a translation of \cite{cite:karatsuba_ant}. |
\newblock This is a translation of \cite{cite:karatsuba_ant}. |
|
|
| \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} |
