Affinely self-generating sets and substitution sequences

Kimberling defined a self-generating set $S$ of integers as follows: 1 is a member of $S$, and if $x$ is a member of $S$, then $2x$ and $4x - 1$ are in $S$. Nothing else is in $S$. As an REU project, I investigated several interesting properties of the Kimberling set, and its resulting sequences reduced modulo m - particularly the case m = 2, which yields the Fibonacci word. Garth and Gouge (2007) proved not only that any of a special class of affinely self-generating sets of integers reduced modulo $m$may be generated by a substitution morphism, but also its characteristic sequence, which turns out to be automatic. The talk will be introductory in nature, and will also discuss some of the open questions posed by Garth and Gouge.

David Failing
David Failing

Lead Data Scientist and Consultant

Mathematics Instructor