Büchi automata are the natural analogue of finite automata in the context of infinite strings (indexed by the natural numbers) on a finite alphabet. We say a subset $X$ of the reals is $r$-regular if there is a Büchi automaton that accepts (one of) the base-$r$ representations of every element in $X$, and rejects the base-$r$ representations of each element in its complement. These sets often exhibit fractal-like behavior—e.g., the Cantor set is 3-regular. There are remarkable connections in logic to Büchi automata, particularly in model theory. In this talk, I will give a characterization of when the expansion of the real ordered additive group by a predicate for a closed r-regular subset of [0,1] is model-theoretically tame (d-minimal, NIP, NTP2). Moreover, I will discuss how this coincides with geometric tameness, namely trivial fractal dimension. This will include a discussion of how the properties of definable sets vary depending on the properties of the Büchi automaton that recognizes the predicate subset.

Bio: Alexi Block Gorman is a model theorist from the United States. Block Gorman completed her thesis at the University of Illinois at Urbana-Champaign under the supervision of Philipp Hieronymi. She is now a Fields postdoctoral fellow. Her research primarily concerns applications of o-minimality and tame geometry.