On Countable SFT Covers of Sparse Multidimensional Shift Spaces

TL;DR

This paper investigates when certain multidimensional shift spaces, specifically gap width shifts, can be covered by countably many configurations in a sofic shift, revealing new conditions for countable coverage.

Contribution

It characterizes one-dimensional gap width shifts with countable covers and demonstrates that many two-dimensional gap width shifts are countably covered.

Findings

Characterization of one-dimensional gap width shifts with countable SFT covers

Identification of a large class of two-dimensional gap width shifts that are countably covered

Contrast with the one-dimensional case where all countable sofic shifts are countably covered

Abstract

A multidimensional sofic shift is called countably covered if it has an SFT cover containing only countably many configurations. In contrast to the one-dimensional setting, not all countable sofic shifts are countably covered. We investigate the existence of countable covers for gap width shifts, where the number of nonzero symbols in a configuration is bounded by a function of the minimum distance between two such symbols. As our main results, we characterize those one-dimensional gap width shifts whose two-dimensional lift is a countably covered sofic shift, and show that a large class of two-dimensional gap width shifts are countably covered.