How small can a set be while containing many configurations? Following up on earlier work of Erdős and Kakutani (Colloq Math 4:195–196, 1957), Máthé (Fund Math 213(3):213–219, 2011) and Molter and Yavicoli (Math Proc Camb Soc 168:57–73, 2018), we address the question in two directions. On one hand, if a subset of the real numbers contains an affine copy of all bounded decreasing sequences, then we show that such a subset must be somewhere dense. On the other hand, given a collection of convergent sequences with prescribed decay, there is a closed and nowhere dense subset of the reals that contains an affine copy of every sequence in that collection.

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关于包含有界递减序列的仿射副本的集合

包含许多配置的集合可以有多小？跟进Erdős和Kakutani（Colloq Math 4：195–196，1957），Máthé（Fund Math 213（3）：213–219，2011）和Molter and Yavicoli（Math Proc Camb Soc 168：57–73）的早期工作（2018年），我们从两个方向解决这个问题。一方面，如果实数的一个子集包含所有有界递减序列的仿射副本，那么我们表明该子集必须在某处密集。另一方面，给定具有规定衰减的收敛序列的集合，实部的封闭且无处密集的子集包含该集合中每个序列的仿射副本。