A paradoxical decomposition of the real line

TL;DR

This paper presents a method to partition the real line into a finite or countable number of subsets that can be reassembled into multiple copies of the line using measure-preserving functions.

Contribution

It introduces a paradoxical decomposition of the real line into subsets that can be reassembled into multiple copies, extending previous results to arbitrary counts.

Findings

Partition into four subsets reassembled into two lines

Extension to 2k pieces forming k lines

Countable partitions forming countably many lines

Abstract

In this paper we demonstrate how to partition the real number line into four subsets which may be reassembled, via "piecewise rigid functions" that preserve Lebesgue measure, into two copies of the line. We then employ a similar process to split the line into $2k$ pieces that yield $k$ copies of the line, or even into countably many subsets to obtain countably many copies of $\mathbb{R}$.