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# 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 Ð?. Ð?. ÐšÐ°Ñ€Ð°Ñ†ÑƒÐ±Ð°. ÐžÑ?Ð½Ð¾Ð²Ñ‹ Ð°Ð½Ð°Ð»Ð¸Ñ‚Ð¸Ñ‡ÐµÑ?ÐºÐ¾Ð¹ Ñ‚ÐµÐ¾Ñ€Ð¸Ð¸ Ñ‡Ð¸Ñ?ÐµÐ». Ð?Ð°ÑƒÐºÐ°, 1983. Zbl 0428.10019. For English translation see [2].
- 2 A. A. Karatsuba. Basic analytic number theory. Springer-Verlag, 1993. Zbl 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. Zbl 0814.11001.

Keywords:

uniform distribution, exponential sums

Related:

UniformlyDistributed

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

11K06*no label found*11K38

*no label found*11L03

*no label found*

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new question: Prime numbers out of sequence by Rubens373

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new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag