论文标题
阿贝尔小组行动可衡量的瓷砖
Measurable tilings by abelian group actions
论文作者
论文摘要
令$ x $为一个量度保留操作$（g，x）\ mapsto g \ cdot x $的度量空间。我们考虑理解可测量瓷砖的结构$ f \ odot a = x $ $ x $ by可测量的瓷砖$ a \ subset x $由有限套件$ f \ subset g $翻译的换档，因此，翻译$ f \ f \ cdot a $，$ f \ in f $ partition $ x $ x $ x $ x $ to null设置。我们建立了一个“扩张引理”的论点，该论点粗略地断言，$ f \ odot a = x $暗示着$ f^r \ odot a = x $，用于大型整数扩张$ r $ r $，并使用它来建立与最近和第二个由四个四个词典建立的结构相似的结构。作为该理论的应用，我们完全对有限产生的阿伯利亚群体的随机整理，这些群体是“ IID因素”的，并表明圆环$ \ mathbb {t}^d $的可测量瓷砖始终可以（实际上（实际上实际上是线性地）变形，并在$ dimemens中的特定范围内，$ DIMENSIDEN $ DIMENSINES $ D = 1（$ DIMES）$ D = 1第一作者康利（Conley）和$ d = 1 $ case中的皮克尔科（Pikhurko）。
Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a "dilation lemma" that asserts, roughly speaking, that $F \odot A = X$ implies $F^r \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are "factors of iid", and show that measurable tilings of a torus $\mathbb{T}^d$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).
