On automatic subsets of the Gaussian integers

Abstract

Suppose that $a$ and $b$ are multiplicatively independent Gaussian integers, that are both of modulus~$\ge \sqrt{5}$. 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.