A Three-Dimensional SFT with Sparse Columns

We construct a nontrivial three-dimensional subshift of finite type whose projective $\mathbb{Z}$-subdynamics, or $\mathbb{Z}$-trace, is 2-sparse, meaning that there are at most two nonzero symbols in any vertical column. The subshift is deterministic in the direction of the subdynamics, so it is topologically conjugate to the set of spacetime diagrams of a partial cellular automaton. We also present a variant of the subshift that is defined by Wang cubes, and one whose alphabet is binary.