# 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 UniformlyDistributed 2013-03-22 14:17:29 2013-03-22 14:17:29 bbukh (348) bbukh (348) 6 bbukh (348) Definition msc 11K06 msc 11K38 equidistributed WeylsCriterion