## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Monday, November 8, 2021 — 10:00 AM EST

**Alexi Block Gorman, Fields Institute**

"Fractal Dimensions and Definability from Büchi Automata"

Büchi automata are the natural extension of finite automata, also called finite-state machines, to a model of computation that accepts infinite-length inputs. 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 of X, and rejects the base-r representations of each element in its complement. We can analogously define r-regular subsets of higher arities, and these sets often exhibit fractal-like behavior--e.g., the Cantor set is 3-regular. There are compelling connections between fractal geometry, r-regular subsets of the reals, and the directed graph structure of the automata that witness regularity. For an r-regular subset of the unit box in n-dimensional Euclidean space, we will describe how to obtain Hausdorff dimension, Box counting dimension, and Hausdorff measure (for the appropriate dimension) in terms of a certain variation of induced sub-automata. We will also see how this gives us a characterization for when reducts of a relevant first-order structure, one expanding the reals as an ordered additive group, have definable unary sets whose Hausdorff dimension and Boxing counting dimension disagree. This is joint work with Christian Schulz.

MC 5417

Event tags

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.