Logic Seminar

The Ki-Linton theorem asserts that the set of base bb normal numbers is a Π03Π30-complete set. The base bb normal numbers can be viewed as the set of generic points for an associated dynamical system. This leads to the question of the complexity of the set of generic points for other numeration/dynamical systems, for example continued fractions, ββ-expansions, Lüroth expansions to name a few. We prove a general result which covers all of these cases, and involves a well-known property in dynamics, a form of the specification property. We then consider differences of these sets. Motivated by the descriptive set theory arguments, we are able to show that the set of continued fraction normal but not base bb normal numbers is a complete D2(Π03)D2(Π30) set. Previously, the best known result was that this set was non-empty (due to Vandehey), and this assumed the generalized Riemann hypothesis. The first part of the work is joint with Mance and Kwietniak, and the second part with Mance and Vandehey.