Quantitative estimates for the size of an intersection of sparse automatic sets

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