Additive systems for $\mathbb{Z}$ are undecidable

TL;DR

This paper investigates additive systems over integers, revealing that determining their properties can be as complex as solving the Collatz conjecture or the halting problem, thus proving undecidability in some cases.

Contribution

It introduces canonical collections for additive systems over integers and links their properties to well-known undecidable problems.

Findings

Some collections' sumset coverage is equivalent to the Collatz conjecture.

Certain well-behaved collections relate to the halting problem for Fractran.

The problem of coverage for these collections is undecidable.

Abstract

What are the collections of sets ${A}_i\subset\mathbb{Z}$ such that any $n\in\mathbb{Z}$ has exactly one representation as $n=a_0+a_1+\dotsb$ with $a_i\in{A}_i$? The answer for $\mathbb{N}_0$ instead of $\mathbb{Z}$ is given by a theorem of de Bruijn. We describe a family of natural candidate collections for $\mathbb{Z}$, which we call canonical collections. Translating the problem into the language of dynamical systems, we show that the question of whether the sumset of a canonical collection covers the entire $\mathbb{Z}$ is difficult: specifically, there is a collection for which this question is equivalent to the Collatz conjecture, and there is a well-behaved family of collections for which this question is equivalent to the universal halting problem for Fractran and is therefore undecidable.