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

A theorem of Cobham says that if k and B are two multiplicatively independent natural numbers then a subset of the natural numbers that is both k- and B -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 B -automatic subset of Nd is finite with size that can be explicitly bounded in terms of data from the automata that accept these sets.(c) 2023 Elsevier B.V. All rights reserved.