Comparison of Coverability and Multi-Scale Coverability in One and Two Dimensions

A word is quasiperiodic (or coverable) if it can be covered with occurrences of another finite word, called its quasiperiod. A word is multi-scale quasiperiodic (or multi-scale coverable) if it has infinitely many different quasiperiods. These notions were previously studied in the domains of text algorithms and combinatorics of right infinite words. We extend them to infinite pictures (two-dimensional words). Then we compare the regularity properties (uniform recurrence, uniform frequencies, topological entropy) of quasiperiodicity with multi-scale quasiperiodicity, and we also compare each of them with its one-dimensional counterpart. We also study which properties of quasiperiods enforce properties on the quasiperiodic words.

💡 Research Summary

The paper investigates the notions of coverability (also called quasiperiodicity) and multi‑scale coverability for infinite words, extending them from the classical one‑dimensional setting to two‑dimensional infinite pictures (functions ℤ² → Σ). A finite block q is a cover of a picture w if every cell of w lies inside some occurrence of q; a picture is coverable when at least one such q exists. Multi‑scale coverability requires infinitely many distinct covers.

After recalling standard regularity concepts—uniform recurrence, uniform frequencies (unique ergodicity), and topological entropy (defined via the growth rate of the block‑complexity function c_w(n,n))—the authors examine how these properties interact with coverability in two dimensions.

Key results on independence:
Using a specific 8 × 1 cover, the authors construct coverable pictures that lack uniform recurrence, lack uniform frequencies, and even have positive topological entropy. This demonstrates that, just as in one dimension, coverability is a purely local condition and does not automatically enforce any of the classical global regularities.

Dependence via cover structure:
The paper then shows that the shape of the cover matters. If a cover q has at least one corner that does not contain a non‑empty “border” (i.e., q does not occupy the full width or full height in that corner), then every q‑coverable picture must have zero topological entropy. The proof analyses how occurrences of q tile the plane, bounding the number of possible n × n blocks by |q|·2^{2n}, which yields polynomial block complexity and thus zero entropy. Conversely, if q has a full‑width or full‑height border, positive entropy pictures can exist, illustrating a clear dependence on the geometry of the cover.

Multi‑scale coverability forces global order:
When infinitely many distinct covers exist, the authors prove that every finite block appears with a well‑defined frequency (uniform frequencies) and that the topological entropy of the picture is necessarily zero. The argument relies on the fact that the infinite hierarchy of increasingly refined covers prevents exponential growth of block complexity. Consequently, multi‑scale coverability implies both uniform frequencies and zero entropy, mirroring known one‑dimensional results but now established for two‑dimensional pictures.

Summary of the relationship tables:
Initially, the authors presented a table indicating that ordinary coverability is independent (⊥) of uniform recurrence, frequencies, and entropy, while multi‑scale coverability implied uniform frequencies (⇒) but left other entries unresolved. Their extended analysis fills the gaps: ordinary coverability remains independent of all three regularities, whereas multi‑scale coverability implies uniform recurrence, uniform frequencies, and zero entropy (all “⇒”).

Open questions and future work:
The paper conjectures that for covers lacking full‑width or full‑height borders but possessing borders in all four corners, the entropy of any q‑coverable picture should also be zero—a case not yet proved. The authors suggest further classification of covers based on border configurations and propose exploring connections with tiling theory and subshift dynamics, where local rules (covers) may or may not generate global order.

Overall contribution:
By systematically extending coverability to two dimensions and dissecting the interplay between local covering rules and global dynamical properties, the paper clarifies when locality yields global regularity and when it does not. It establishes that multi‑scale coverability is a robust notion that enforces uniform recurrence, uniform frequencies, and zero entropy, while ordinary coverability alone does not. These insights enrich the combinatorics on words, symbolic dynamics, and theoretical computer science, offering new tools for analyzing two‑dimensional symbolic systems and tilings.