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 isuniformly 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 measurespace $(X,𝒜,\mu )$ is uniformly distributed with respectto a family of sets $ℱ\subset 𝒜$ if

$$\underset{N\to \mathrm{\infty }}{lim}\frac{\mathrm{card}\{n\in [1..N]:u_n\in S\}}{N}=\frac{\mu (S)}{\mu (X)}\mathit{\hspace{1em}\hspace{1em}}\text{for every }S\in ℱ.$$