2007 Student Research Conference:

20th Annual Student Research Conference

20th Annual Student Research Conference

**Affinely Self-Generating Sets and Morphisms**

Adam C. Gouge

Dr. David Garth, Faculty Mentor

Kimberling defined a self-generating set *S* of integers as follows. Assume 1 is a member of *S*, and if *x* is in *S*, then 2*x* and 4*x*-1 are in *S*. We study similar self-generating sets of integers whose generating functions come from a class of affine functions for which the coefficients of *x* are powers of a fixed base. We prove that for any positive integer *m*, the resulting sequence, reduced modulo *m*, is the image of an infinite word that is the fixed point of a morphism over a finite alphabet. We also prove that the resulting characteristic sequence of *S* is the image of the fixed point of a morphism of constant length, and is therefore automatic.

**Keywords:** mathematics, self-generating sets, sequences, affine functions, morphisms, modulo arithmetic, missing blocks, Fibonacci numbers

**Topic(s):**Mathematics

**Presentation Type:** Poster

**Session:** 5-1

**Location:** OP Lobby

**Time:** 4:15 pm