Lecture Course II:

There are various ways of expanding real numbers and vectors into convergent series with a certain `digit set'. The most common method is to use decimal (or binary) expansions where the digits are independent and identically distributed. However, in some problems it is more natural to look at expansions in a base which is not an integer. The traditional way of doing so is via applying the greedy alrorithm, however by far not the only one. This course is aimed at describing dynamical, arithmetic and combinatorial aspects of expansions in non-integer bases as well as their applications to measure theory and fractal geometry.

SYLLABUS:

• Expansions in non-integer bases for reals and vectors: different algorithms of obtaining digits.
• Combinatorics, arithmetic and dynamics related to such expansions.
• Application: Bernoulli convolutions.
• Two-dimensional expansions and fractals

Lecture Notes	Exercises 1	Exercises 2

BACKGROUND READING:

• K. Falconer `Techniques in Fractal Geometry'
• N. Sidorov, `Arithmetic Dynamics'
• B. Solomyak, `Notes on Bernoulli Convolutions'