A note on syndeticity, recognizable sets and Cobham's theorem

TL;DR

This paper provides an alternative proof that infinite sets recognizable in two multiplicatively independent bases are syndetic, aiding the classical proof of Cobham's theorem and enhancing understanding of recognizable sets.

Contribution

It offers a new, accessible proof of a key step in Cobham's theorem, connecting recognizability and syndeticity for sets in different bases.

Findings

Infinite p- and q-recognizable sets are syndetic for multiplicatively independent p and q

Supports the classical proof of Cobham's theorem

Completes previous work to make the proof more accessible

Abstract

In this note, we give an alternative proof of the following result. Let p, q >= 2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is syndetic. Notice that this result is needed in the classical proof of the celebrated Cobham?s theorem. Therefore the aim of this paper is to complete [13] and [1] to obtain an accessible proof of Cobham?s theorem.