Quasiperiodicity and non-computability in tilings

TL;DR

This paper constructs a tile set that produces tilings which are both quasiperiodic and non-recursive, advancing understanding of the interplay between combinatorial and algorithmic properties in tilings.

Contribution

It introduces a novel fixed point construction that enforces local regularity for quasiperiodicity and shows how effectively closed sets can be transformed into tile sets with specific Turing degree properties.

Findings

Existence of tile sets with quasiperiodic and non-recursive tilings

Enhanced fixed point construction for local regularity

Transformation of effectively closed sets into tile sets with controlled Turing degrees

Abstract

We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the fixed point construction; we improve this general technique and make it enforce the property of local regularity of tilings needed for quasiperiodicity. We prove also a stronger result: any effectively closed set can be recursively transformed into a tile set so that the Turing degrees of the resulted tilings consists exactly of the upper cone based on the Turing degrees of the later.