uniformly distributed

Let $\{u_{n}\}$ be a sequence of real numbers. For $0\leq\alpha<\beta\leq 1$ put

Z(N,\alpha,\beta)=\operatorname{card}\{n\in[1..N]:\alpha\leq(u_{n}\bmod 1)<% \beta\}.

The sequence $\{u_{n}\}$ is uniformly distributed modulo $1$ if

\lim_{N\to\infty}\frac{1}{N}Z(N,\alpha,\beta)=\beta-\alpha

for all $0\leq\alpha<\beta\leq 1$ . 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

\lim_{N\to\infty}\frac{\operatorname{card}\{n\in[1..N]:u_{n}\in S\}}{N}=\frac{% \mu(S)}{\mu(X)}\qquad\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
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