On the Number of Kernel Elements of Automatic Sequences

We consider bi-infinite k-automatic sequences, i.e., maps \(f:\mathbb{Z} \to \mathcal{R}\) with values in a finite ring \(\mathcal{R}\). We study the dependence of the number of kernel elements on the particular choice of a k-residue set. We establish several upper estimates for the number of kernel elements, in particular, for decimation invariant- and periodic sequences.