# On Sets Containing an Affine Copy of Bounded Decreasing Sequences

## TL;DR

This paper investigates the minimal size and density properties of sets containing affine copies of bounded decreasing sequences, showing such sets must be dense or can be constructed as nowhere dense, depending on the context.

## Contribution

It establishes conditions under which sets containing affine copies of all bounded decreasing sequences must be dense, and constructs nowhere dense sets for specific collections of sequences.

## Key findings

- Sets containing all bounded decreasing sequences are necessarily somewhere dense.
- Existence of closed, nowhere dense sets containing affine copies of specific convergent sequences.
- Characterization of set size and density constraints for containing affine copies of sequences.

## Abstract

How small can a set be while containing many configurations? Following up on earlier work of Erd\H os and Kakutani \cite{MR0089886}, M\'ath\'e \cite{MR2822418} and Molter and Yavicoli \cite{Molter}, 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 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.