Dynamical Systems in Number Theory

Many problems in number theory may be stated as problems relating to the uniform distribution of certain numerical sequences. Recall that the sequence x₁, x₂,...., 0 ≤ x _n ≤ 1, is uniformly distributed on the closed interval [0, 1], if for any function f ∈ C([0, 1]) we have the relation $$ \mathop{{\lim }}\limits_{{n \to \infty }} \frac{1}{n}\sum\limits_{{k = 1}}^{n} {f({x_{k}}) = \int_{0}^{1} {f(x)dx} } $$. Similarly, we can define the uniform distribution on an arbitrary closed interval [a, b], a < b.