Title: Descriptive complexity in number theory and dynamics
Speaker: Bill Mance (Adam Mickiewicz University in Poznań)
Abstract: Informally, a real number is normal in base $b$ if in its $b$-ary expansion, all digits and blocks of digits occur as often as one would expect them to, uniformly at random. We will denote the set of numbers normal in base $b$ by $\mathcal{N}(b)$. Kechris asked several questions involving descriptive complexity of sets of normal numbers. The first of these was resolved in 1994 when Ki and Linton proved that $\mathcal{N}(b)$ is $\boldsymbol{\Pi}_3^0$-complete. Further questions were resolved by Becher, Heiber, and Slaman who showed that $\bigcap_{b=2}^\infty \mathcal{N}(b)$ is $\boldsymbol{\Pi}_3^0$-complete and that $\bigcup_{b=2}^\infty \mathcal{N}(b)$ is $\boldsymbol{\Sigma}_4^0$-complete. Many of the techniques used in these proofs can be used elsewhere. We will discuss recent results where similar techniques were applied to solve a problem of Sharkovsky and Sivak and a question of Kolyada, Misiurewicz, and Snoha. Furthermore, we will discuss a recent result where the set of numbers that are continued fraction normal, but not normal in any base $b$, was shown to be complete at the expected level of $D_2(\boldsymbol{\Pi}_3^0)$. An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.