从维数下降到非周期序
From Dimension Drop to Aperiodic Order
- University of St Andrews（圣安德鲁斯大学）
- Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文对双几何级数可达集族给出完整拓扑分类，其二元分类诱导出格点上的非周期双色铺砌，并揭示维数下降现象，证明该铺砌是四个算术定义状态上替换铺砌的因子。
AI中文摘要：
我们考虑集合族 $$ E(x,y)=\left\{\sum_{n=1}^{\infty}\frac{\varepsilon_n}{4^n}: (\varepsilon_n) \in \{0,x,y,x+y\}^{\mathbb{N}}\right\} $$ 其中 $(x,y) \in \mathbb{N}^2$。该族可通过三个视角来观察：(a) 作为齐次自相似集；(b) 作为双几何级数的可达集；或 (c) 作为四角康托尔集的“有理”正交投影集。我们综合这三个视角，对 $(x,y) \in \mathbb{N}^2$ 时 $E(x,y)$ 的拓扑分类给出了完整刻画。接下来，我们根据 $E(x,y)$ 是否具有内部，将此拓扑分类压缩为二元分类。当可视化这一二元分类时，它揭示了格点 $\mathbb N^2$ 的一个双色铺砌 $T$，该铺砌尽管明显具有结构，却被证明是非周期的；事实上，我们证明它没有非平凡的平移对称性。由于我们模型的刚性，这一相同的二元分类同时捕获了若干二分法。最值得注意的是，当通过自相似集理论来审视族 $\{E(x,y)\}_{(x,y) \in \mathbb{N}^2}$ 时，$T$ 可被视为描述了族内维数下降的出现。最后，我们考察了铺砌非周期序背后的机制。通过考虑铺砌的数论性质，我们刻画了其替换结构，并发现 $T$ 是四个“隐藏”的算术定义状态上的替换铺砌的一个因子。
英文摘要：
We consider the parametrised family of sets $$ E(x,y)=\left\{\sum_{n=1}^{\infty}\frac{\varepsilon_n}{4^n}: (\varepsilon_n) \in \{0,x,y,x+y\}^{\mathbb{N}}\right\} $$ for $(x,y) \in \mathbb{N}^2$. This family can be viewed through three lenses: (a) as homogeneous self-similar sets; (b) as achievement sets of bi-geometric series; or (c) as the set of `rational' orthogonal projections of the four-corner Cantor set. We synthesise these three perspectives to obtain a complete topological classification of $E(x,y)$ for $(x,y) \in \mathbb{N}^2$. Next, we collapse this topological classification to a binary one according to whether or not $E(x,y)$ has interior. When this binary classification is visualised, it reveals a two-colour tiling $T$ of the lattice $\mathbb N^2$, which, despite being visibly structured, turns out to be aperiodic; indeed, we prove it has no non-trivial translational symmetries. Due to the rigidity of our model, this same binary classification simultaneously captures several dichotomies. Most notably, when the family $\{E(x,y)\}_{(x,y) \in \mathbb{N}^2}$ is viewed through the theory of self-similar sets, $T$ can be seen to describe the emergence of dimension drop within the family. Finally we examine the mechanism underlying the tiling's aperiodic order. By considering the number-theoretic properties of the tiling, we characterise its substitution structure, and discover that $T$ is a factor of a substitution tiling on four ``hidden'' arithmetically defined states.
