Abstract: This theory grew out of an example in a historical paper by A. Rényi in 1956, illustrating the use of ergodic theory in investigating the frequency of digits, a problem going back to a famous theorem of Borel. Subsequent research revealed many hitherto hidden connections of his example to number theory, topology, fractals, symbolic dynamics and probability. The theory got a new impetus with the discovery in 1990 of P. Erdős and his collaborators of strange and unexpected properties of unique expansions. This led to hundreds of papers in the past thirty years. We propose an introduction to this subject, which is still rich in open problems.
The plan for our course is the following:
- Definition of expansions, examples illustrating the differences between integer and non-integer bases.
- Lexicographic characterizations.
- The number of expansions.
- Univoque bases.
- Topological description of the set of numbers having a unique expansion.
- Hausdorff dimension of the preceeding set.
- Multiple expansions.
For additional information, send email to marco.pedicini@uniroma3.it
Il corso avrà luogo in presenza presso il Dipartimento di Matematica e Fisica, L.go S. L. Murialdo 1 - Aula 311