# Weyl’s criterion

Let $\{u_{n}\}$ be a sequence of real numbers. Then $\{u_{n}\}$ is uniformly distributed modulo $1$ if and only if

 $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}e(ku_{n})=0$

for every nonzero integer $k$, where $e(x)=\exp(2\pi ix)$.

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:

 $\left\lvert\sum_{n=1}^{N}e(knx)\right\rvert=\left\lvert\frac{e(k(N+1)x)-e(kx)}% {e(kx)-1}\right\rvert\leq\frac{2}{\left\lvert\,e(kx)-1\right\rvert}=O_{k}(1)$

because the irrationality of $x$ implies $e(kx)\neq 1$.

## References

• 1 Ã?. Ã?. ÃÅ¡ÃÂ°Ãâ¬ÃÂ°Ãâ ÃÆÃÂ±ÃÂ°. ÃÅ¾Ã?ÃÂ½ÃÂ¾ÃÂ²Ãâ¹ ÃÂ°ÃÂ½ÃÂ°ÃÂ»ÃÂ¸ÃâÃÂ¸Ãâ¡ÃÂµÃ?ÃÂºÃÂ¾ÃÂ¹ ÃâÃÂµÃÂ¾Ãâ¬ÃÂ¸ÃÂ¸ Ãâ¡ÃÂ¸Ã?ÃÂµÃÂ». Ã?ÃÂ°ÃÆÃÂºÃÂ°, 1983. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0428.10019Zbl 0428.10019.
• 2 A. A. Karatsuba. 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 WeylsCriterion 2013-03-22 14:17:31 2013-03-22 14:17:31 bbukh (348) bbukh (348) 7 bbukh (348) Theorem msc 11K06 msc 11K38 msc 11L03 UniformlyDistributed