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

Abstract

A theorem of Cobham says that if k and ℓ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both k- and ℓ-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 Nd and a sparse ℓ-automatic subset of Nd is finite with size that can be explicitly bounded in terms of data from the automata that accept these sets.