Abstract: Ideas from dynamics have had a certain degree of success in problems of diophantine approximation. After a brief discussion of some successes, we shall discuss a "dynamical" approach a longstanding open question: Does |(3/2)^n-m|<\epsilon have a solution in natural numbers m,n? (Of course, you can replace 3/2 by any p/q>1.)

We will be led to a variety of topics: cellular automata, Hausdorff dimension, algebraic functions over finite fields, intersection of "nice" fractals, etc.

The lecture will pose many problems and provide few solutions.