Weyl’s criterion
Let $\{{u}_{n}\}$ be a sequence of real numbers. Then $\{{u}_{n}\}$ is uniformly distributed modulo $1$ if and only if
$$\underset{N\to \mathrm{\infty}}{lim}\frac{1}{N}\sum _{n=1}^{N}e(k{u}_{n})=0$$ |
for every nonzero integer $k$, where $e(x)=\mathrm{exp}(2\pi ix)$.
Weyl’s criterion reduces the problem of uniform distribution^{} of sequences to the problem of estimating certain exponential sums. Whereas the problem of estimating a family of exponential sums might seem harder at first, the exponential map has the multiplicative property which often makes the problem easier.
Example: If $x$ is irrational, then the sequence $\{nx\}$ is uniformly distributed modulo $1$. Proof:
$$\left|\sum _{n=1}^{N}e(knx)\right|=\left|\frac{e(k(N+1)x)-e(kx)}{e(kx)-1}\right|\le \frac{2}{\left|e(kx)-1\right|}={O}_{k}(1)$$ |
because the irrationality of $x$ implies $e(kx)\ne 1$.
References
- 1 Ã?. Ã?. ÃÅ¡ÃÂ°Ãâ¬ÃÂ°Ãâ ÃÆÃÂ±ÃÂ°. ÃÅ¾Ã?ÃÂ½ÃÂ¾ÃÂ²Ãâ¹ ÃÂ°ÃÂ½ÃÂ°ÃÂ»ÃÂ¸ÃâÃÂ¸Ãâ¡ÃÂµÃ?ÃÂºÃÂ¾ÃÂ¹ ÃâÃÂµÃÂ¾Ãâ¬ÃÂ¸ÃÂ¸ Ãâ¡ÃÂ¸Ã?ÃÂµÃÂ». Ã?ÃÂ°ÃÆÃÂºÃÂ°, 1983. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0428.10019Zbl 0428.10019. For English translation^{} see [2].
- 2 A. A. Karatsuba. Basic analytic number theory^{}. Springer-Verlag, 1993. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0767.11001Zbl 0767.11001.
- 3 Hugh L. Montgomery. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, volume 84 of Regional Conference Series in Mathematics. AMS, 1994. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0814.11001Zbl 0814.11001.
Title | Weyl’s criterion |
---|---|
Canonical name | WeylsCriterion |
Date of creation | 2013-03-22 14:17:31 |
Last modified on | 2013-03-22 14:17:31 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 7 |
Author | bbukh (348) |
Entry type | Theorem |
Classification | msc 11K06 |
Classification | msc 11K38 |
Classification | msc 11L03 |
Related topic | UniformlyDistributed |