关于具有全仿射跨度的非凸窗口的尼瓦特猜想的一个反例
A Counterexample to Nivat's Conjecture for a Non-Convex Window of Full Affine Span
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中文总结 AI 辅助
构造基数为8且具全仿射跨度的精确簇\(F\)及\(F\)-平铺\(T\)，给出非凸窗口尼瓦特猜想的非退化反例否定卡里和穆托问题；还证明基数为质数平方的\(F\)的\(F\)-平铺轨道闭包有1-周期平铺，表明少于8个单元簇一般无此现象。
中文摘要 AI 辅助
我们构造了一个基数为8且具有全仿射跨度的精确簇\(F\subseteq\mathbb{Z}^2\)以及一个\(F\)-平铺\(T\)，使得\(T\)在\(\{0,1\}^{\mathbb{Z}^2}\)中的轨道闭包不包含1-周期的\(F\)-平铺。由于每个\(F\)-平铺相对于窗口\(\bar F = \{ -a : a \in F\}\)是低复杂度配置，这在强意义上为非凸窗口的尼瓦特猜想提供了一个“非退化”反例。这否定地回答了卡里和穆托（2023年）的一个问题，即每个这样的反例是否一定是退化的，即探测窗口包含在一个适当的有限指数子格的陪集中。我们用一个正面结果补充这一点：对于每个具有全仿射跨度且基数为质数平方的精确簇\(F\)，每个\(F\)-平铺在其轨道闭包中都有一个1-周期的\(F\)-平铺。结合塞格迪定理，即由质数基数簇进行的每个平铺都是1-周期的，这表明少于8个单元的簇不会出现这种现象，基数为6的情况可能除外，我们对此未解决。
英文摘要
We construct an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality $8$ with full affine span, together with an $F$-tiling $T$, such that the orbit closure of $T$ in $\{0,1\}^{\mathbb{Z}^2}$ does not contain a $1$-periodic $F$-tiling. Since every $F$-tiling is a low-complexity configuration with respect to the window $\bar F := \{-a : a \in F\}$, this supplies a "non-degenerate" counterexample, in a strong sense, to Nivat's conjecture for non-convex windows. This answers, in the negative, a question of Kari and Moutot (2023) whether every such counterexample must be degenerate, in the sense that the probing window is contained in a coset of a proper finite-index sublattice. We complement this with a positive result: for every exact cluster $F$ of full affine span whose cardinality is the square of a prime, every $F$-tiling has a $1$-periodic $F$-tiling in its orbit closure. Together with Szegedy's theorem that every tiling by a cluster of prime cardinality is $1$-periodic, this shows that no cluster of fewer than $8$ cells can exhibit the phenomenon, with the possible exception of cardinality $6$, which we leave open.
