On automatic subsets of the Gaussian integers

TL;DR

This paper proves the existence of subsets of Gaussian integers that are automatic with respect to one Gaussian integer but not another, resolving a previously open problem in the theory of automatic sequences.

Contribution

It establishes that for multiplicatively independent Gaussian integers of modulus at least √5, there exist subsets that are automatic for one but not the other, answering a longstanding question.

Findings

Existence of $a$-automatic but not $b$-automatic subsets for certain Gaussian integers.

Resolution of a problem posed by Allouche et al.

Advancement in understanding automata over Gaussian integers.

Abstract

Suppose that $a$ and $b$ are multiplicatively independent Gaussian integers, that are both of modulus~$\geq \sqrt 5$. We prove that there exist a $X\subset \mathbb Z[i]$ which is $a$-automatic but not $b$-automatic. This settles a problem of Allouche, Cateland, Gilbert, Peitgen, Shallit, and Skordev.