TITLE: Descriptive complexity in number theory and dynamics
SPEAKER: William Mance, University of Adam Mickiewicz in Poznan, Poland
DATE: 16:00-18:00, 1 March 2024
VENUE: Sendai Logic Seminar, The Mathematical Institute,
Graduate School of Science, Tohoku University
Room 801, Science Complex A
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.
Contact: Yohji Akama
The Mathematical Institute,
Graduate School of Science, Tohoku University
yoji.akama.e8@tohoku.ac.jp