On automatic subsets of the Gaussian integers
  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. 
Wieb Bosma, Robbert Fokkink, Thijmen Krebs
cs.FL