Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons P and Q are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice $${\mathbb {Z}}^2$$ Z 2 ) from P to Q . We develop an invariant for a particular version of this notion called rational finite discrete equidecomposability . We construct triangles that are Ehrhart equivalent but not rationally finitely discretely equidecomposable, thus providing a partial negative answer to a question of Haase–McAllister on whether Ehrhart equivalence implies discrete equidecomposability. Surprisingly, if we delete an edge from each of these triangles, there exists an infinite rational discrete equidecomposability relation between them. Our final section addresses the topic of infinite equidecomposability with concrete examples and a potential setting for further investigation of this phenomenon.

中文翻译：

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多边形的离散可分解性和 Ehrhart 理论

受 Ehrhart 理论问题的启发，我们提出了关于离散等分解性的新结果。如果存在分段仿射单模双射（等效地，保留整数格 $${\mathbb {Z}}^2$$ Z 2 的分段仿射线性双射），则称两个有理多边形 P 和 Q 是离散等分解的) 从 P 到 Q 。我们为这个概念的一个特定版本开发了一个不变量，称为有理有限离散 equidecomposability 。我们构造了 Ehrhart 等价但不是合理有限离散等分可分解的三角形，从而为 Haase-McAllister 关于 Ehrhart 等价是否意味着离散等分的问题提供了部分否定的答案。令人惊讶的是，如果我们从每个三角形中删除一条边，它们之间存在着无穷理性离散等分解性关系。我们的最后一节通过具体的例子和进一步研究这种现象的潜在环境来讨论无限等分解性的主题。