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

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.