2025/04/29 by Herva, Pyry, Kari, Jarkko
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.20709
We study complexity and periodicity of Delone sets by applying an algebraic approach to multidimensional symbolic dynamics. In this algebraic approach, ℤd-configurations c: ℤd → A for a finite set A ⊆ ℂ and finite ℤd-patterns are regarded as formal power series and Laurent polynomials, respectively. In this paper we study also functions c: ℝd → A where A is as above. These functions are called ℝd-configurations. Any Delone set may be regarded as an ℝd-configuration by simply presenting it as its indicator function. Conversely, any ℝd-configuration whose support (that is, the set of cells for which the configuration gets non-zero values) is a Delone set can be seen as a colored Delone set. We generalize the concept of annihilators and periodizers of ℤd-configurations for ℝd-configurations. We show that if an ℝd-configuration has a non-trivial annihilator, that is, if a linear combination of some finitely many of its translations is the zero function, then it has an annihilator of a particular form. Moreover, we show that ℝd-configurations with integer coefficients that have non-trivial annihilators are sums of finitely many periodic functions c1,…,cm: ℝd → ℤ. Also, ℝd-pattern complexity is studied alongside with the classical patch-complexity of Delone sets. We point out the fact that sufficiently low ℝd-pattern complexity of an ℝd-configuration implies the existence of non-trivial annihilators. Moreover, it is shown that if a Meyer set has sufficiently slow patch-complexity growth, then it has a non-trivial annihilator. Finally, a condition for forced periodicity of colored Delone sets of finite local complexity is provided.