To provide an introduction to uniform distribution modulo one as a meeting ground for other branches of pure mathematics such as topology, analysis and number theory.

To give an idea of how these subjects work together in this context to shape modern analysis by focusing on one particular rich context.

After completing the module students should be able to:

understand completions and irrationality;

understand diophantine approximation and its relation to uniform distribution;

appreciate that analysis has a complex unity and to have a feel for basic computations in analysis.

calculate rational approximations to real and p adic numbers and to put this to use in number theoretic situations.

calculate approximations to functions from families of simpler functions e.g. continuous functions by polynomials and step functions by trigonometric sums.

work with basic tools from analysis, like Fourier series and continuous functions to prove d istributional properties of sequences of numbers.

Metric spaces, compactness, completeness and connectedness with an emphasis on completions (3 lectures).

Construction of the real line (2 lectures).

The p - adic numbers (4 lectures).

Diophantine approximation, Dirichlet''s theorem, Kronecker''s theorem, Euclid''s algorithm, continued fractions, Pell''s equation and the continued fraction map, and the Gauss measure (6 lectures) .

Continuous functions, Riemann integrable functions and the Stone Weierstrass theorem (4 lectures) .

Uniform distribution, Weyl''s criterion, Van der Korput''s inequality, Weyl''s theorem on the uniform distribution of polynomials and Benford''s law (6 lectures).

Normal numbers, null sets, Borel''s theorem and Champaneau''s number (3 lectures).

Uniform distribution and numerical integration (3 lectures).

Random number generators and their limits (2 lectures).

Discussion of Birkhoff''s ergodic theorem without proof. Application to normality and continued fractions (3 lectures).