Descriptive complexity in numeration systems
Steve Jackson (University of North Texas)
Abstract: Descriptive set theory gives a means of calibrating the complexity of sets, and we focus on some sets occurring in numerations systems. Also, the descriptive complexity of the difference of two sets gives a notion of the logical independence of the sets. A classic result of Ki and Linton says that the set of normal numbers for a given base is a $\boldsymbol{\Pi}^0_3$ complete set. In work with Airey, Kwietniak, and Mance we extend to other numerations systems such as continued fractions, $\beta$-expansions, and GLS expansions. In work with Mance and Vandehey we show that the numbers which are continued fraction normal but not base $b$ normal is complete at the expected level of $D_2(\boldsymbol{\Pi}^0_3)$. An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.
dynamical systemsnumber theory
Audience: researchers in the topic
Series comments: Description: Online seminar on numeration systems and related topics
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| Organizers: | Shigeki Akiyama, Ayreena Bakhtawar, Karma Dajani, Kevin Hare, Hajime Kaneko, Niels Langeveld, Lingmin Liao, Wolfgang Steiner* |
| *contact for this listing |