1. Quantitative estimates for the size of an intersection of sparse automatic sets
- Author
-
Albayrak, Seda and Bell, Jason
- Subjects
FOS: Computer and information sciences ,Mathematics - Number Theory ,Formal Languages and Automata Theory (cs.FL) ,FOS: Mathematics ,Computer Science - Formal Languages and Automata Theory ,Number Theory (math.NT) ,68Q45, 11B85 - Abstract
A theorem of Cobham says that if $k$ and $\ell$ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both $k$- and $\ell$-automatic is eventually periodic. A multidimensional extension was later given by Semenov. In this paper, we give a quantitative version of the Cobham-Semenov theorem for sparse automatic sets, showing that the intersection of a sparse $k$-automatic subset of $\mathbb{N}^d$ and a sparse $\ell$-automatic subset of $\mathbb{N}^d$ is finite with size that can be explicitly bounded in terms of data from the automata that accept these sets., 14 pages
- Published
- 2023