We give a necessary and sufficient condition on a d -dimensional affine subspace of $${\mathbb {R}}^n$$ R n to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of coincidence and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns.

中文翻译：

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由模式定义的规范投影平铺

我们给出了 $${\mathbb {R}}^n$$ R n 的 ad 维仿射子空间的一个充要条件，其特征为一组在其数字化中禁止出现的有限模式。这也可以根据规范投影平铺的局部规则或有限类型的子移位来说明。这提供了仿射子空间的代数性质与其数字化组合学之间的联系。该条件依赖于巧合的概念并且可以被有效地检查。作为推论，我们得到只有代数子空间可以用模式来表征。