Seas of squares with sizes from a $\Pi^0_1$ set

TL;DR

This paper proves that certain two-dimensional subshifts composed of squares with sizes from a $ ext{Pi}^0_1$ set are sofic, extending previous tiling constructions to a broader class of shifts.

Contribution

It extends the self-similar tiling construction to show that $S$-square shifts and effectively closed subshifts of the distinct-square shift are sofic.

Findings

$S$-square shifts are sofic for $ ext{Pi}^0_1$ sets.

Effectively closed subshifts of the distinct-square shift are sofic.

Generalizes tiling methods to broader classes of subshifts.

Abstract

For each $\Pi^0_1$ $S\subseteq \mathbb{N}$, let the $S$-square shift be the two-dimensional subshift on the alphabet $\{0,1\}$ whose elements consist of squares of 1s of various sizes on a background of 0s, where the side length of each square is in $S$. Similarly, let the distinct-square shift consist of seas of squares such that no two finite squares have the same size. Extending the self-similar Turing machine tiling construction of Durand, Romashchenko and Shen, we show that if $X$ is an $S$-square shift or any effectively closed subshift of the distinct square shift, then $X$ is sofic.