Given two distinct integers \(m_{1}, m_{2}\geqslant 2\) , we set \(\alpha _{1}=[0;\overline{1,m_{1}}]\) and \(\alpha _{2}=[0;\overline{1,m_{2}}]\) . We denote by \(S_{\alpha _{1}}(n)\) and \(S_{\alpha _{2}}(n)\) the sum-of-digits functions in the Ostrowski \(\alpha _{1}\) - and \(\alpha _{2}\) -representations of a given positive integer n. In this paper, our goal is to study the arithmetic structure of sets characterized by the Ostrowski sum-of-digits function.
