Let $\{u_n\}$ be a sequence of real numbers. For $0\leq\alpha<\beta\leq 1$ put \begin{equation*} Z(N,\alpha,\beta)=\operatorname{card}\{n\in[1..N] : \alpha \leq (u_n \bmod 1)< \beta \}. \end{equation*}The sequence $\{u_n\}$ is uniformly distributed modulo $1$ if \begin{equation*}\label{eq:modcond} \lim_{N\to\infty} \frac{1}{N} Z(N,\alpha,\beta)=\beta-\alpha \end{equation*}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 \begin{equation*} \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}. \end{equation*}

References

1

William Chen.
Lectures on irregularities of point distribution.
Available at http://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.
Zbl 0814.11001.