Additive systems for $\mathbb{Z}$ are undecidable
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913023277400064 |
|---|---|
| author | Zabolotskii, Andrei |
| author_facet | Zabolotskii, Andrei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive systems for $\mathbb{Z}$ are undecidable Zabolotskii, Andrei Combinatorics Logic in Computer Science 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. |
| title | Additive systems for $\mathbb{Z}$ are undecidable |
| topic | Combinatorics Logic in Computer Science |
| url | https://arxiv.org/abs/2508.17285 |