Decomposing the real line into Borel sets closed under addition

TL;DR

This paper investigates how the real line can be partitioned into Borel sets closed under addition, revealing that such partitions are either very small or uncountably large, with some results undecidable within standard set theory.

Contribution

It establishes bounds on the number of Borel sets closed under addition that partition the real line and explores the undecidability of these partitions in ZFC and related theories.

Findings

Number of such Borel partitions is at most 3 or uncountable.

The existence of certain partitions is undecidable in ZFC.

Various versions of the problem are examined, including dropping Borelness and changing operations.

Abstract

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in $ZFC$ and even in the theory $ZFC + \mathfrak{c} = \omega_2$ if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition $(0,\infty)$, and so on.