Tom Kempton

Measures on distances between polynomials.

Let beta be an algebraic integer. We are interested in the distances between 0-1 polynomials in beta. In this talk we will mainly focus on the case of the golden mean, as a sample question we might ask:

Given words a_0,..., a_n and b_0,...., b_n with each a_i and b_i picked independently from {0,1} with equal probability, what is the probability that the difference

\sum_{i=0}^n a_i\phi^i -\sum_{i=0}^n b_i\phi^i

is equal to one? What is the probability that it is equal to d for some other distance d? How does this change as n tends to infinity?

We will explore answers to these questions and show how dynamics comes in to play. We will also look at why these questions have relevance to fractal geometry. This is joint work with Alex Batsis (Manchester).