关于R^3中的随机紧集、等分解和扩张域
On random compact sets, equidecomposition, and domains of expansion in R^3
摘要：
我们研究R^3的随机紧子集，它们可以被描述为“随机门格尔海绵”。我们用这些随机集构造了R^3中的一对紧致集A和B，它们具有相同的正测度，使得A可以被B的100个平移所覆盖，B可以被A的100个平移所覆盖，但A和B不是等分解的。此外，我们构造了正测度R^3的紧致子集的第一个例子，该子集不是展开域。这回答了Adrian Ioana的问题。
英文摘要：
We study random compact subsets of R^3 which can be described as "random Menger sponges". We use those random sets to construct a pair of compact sets A and B in R^3 which are of the same positive measure, such that A can be covered by finitely many translates of B, B can be covered by finitely many translates of A, and yet A and B are not equidecomposable. Furthermore, we construct the first example of a compact subset of R^3 of positive measure which is not a domain of expansion. This answers a question of Adrian Ioana.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
