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

A theorem of Cobham says that if $k$ and $\mathrm{\ell }$ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both $k$- and $\mathrm{\ell }$-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 $\mathrm{\ell }$-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.