uniformly distributed
Let $\{{u}_{n}\}$ be a sequence of real numbers. For $$ put
$$ |
The sequence $\{{u}_{n}\}$ is uniformly distributed modulo $\mathrm{1}$ if
$$\underset{N\to \mathrm{\infty}}{lim}\frac{1}{N}Z(N,\alpha ,\beta )=\beta -\alpha $$ |
for all $$. In other words a sequence is uniformly distributed modulo $1$ if each subinterval of $[0,1]$ gets its “fair share” of fractional parts of $\{{u}_{n}\}$.
More generally, a sequence $\{{u}_{n}\}$ of points in a finite measure space $(X,\mathcal{A},\mu )$ is uniformly distributed with respect to a family of sets $\mathcal{F}\subset \mathcal{A}$ if
$$\underset{N\to \mathrm{\infty}}{lim}\frac{\mathrm{card}\{n\in [1..N]:{u}_{n}\in S\}}{N}=\frac{\mu (S)}{\mu (X)}\mathit{\hspace{1em}\hspace{1em}}\text{for every}S\in \mathcal{F}.$$ |
References
- 1 William Chen. Lectures on irregularities of point distribution. Available at http://www.maths.mq.edu.au/ wchen/ln.htmlhttp://www.maths.mq.edu.au/ wchen/ln.html, 2000.
- 2 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 | uniformly distributed |
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Canonical name | UniformlyDistributed |
Date of creation | 2013-03-22 14:17:29 |
Last modified on | 2013-03-22 14:17:29 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 6 |
Author | bbukh (348) |
Entry type | Definition |
Classification | msc 11K06 |
Classification | msc 11K38 |
Synonym | equidistributed |
Related topic | WeylsCriterion |