Towards a theorem of Cobham for Gaussian integers
A celebrated theorem of \textit{A. Cobham} asserts that if a set of integers is both \(p\)- and \(q\)-recognizable, where \(p\) and \(q\) are two integers \(\geq 2\) that are multiplicatively independent, then this set is the union of a finite set and of finitely many arithmetic progressions [Math. Syst. Theory 3, 186--192 (1969; Zbl 0179.02501)]. The authors give a first step for a similar result where \(p\) and \(q\) are replaced by \(-p+i\) and \(-q+i\) \((p>0\), \(q>0\), \(p\neq q\), \(p\) and \(q\) integers) and the set of integers is replaced by a set of Gauss integers: they prove that, assuming the four exponentials conjecture, such a set is syndetic (i.e., without ``big holes''). Please note that Reference [11] appeared with a slightly different title [\textit{P. Mihăilescu}, ``Primary cyclotomic units and a proof of Catalan's conjecture'', J. Reine Angew. Math. 572, 167--195 (2004; Zbl 1067.11017)].
scientific article; zbMATH DE number 2174591