お誘い合わせの上ご来訪ください。 正規数の背景の説明も丁寧にするようにお願いしています。

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 [email protected]