On \(U\)-sets for multiplicative systems
Let \(\psi_0,\psi_1,\dots\) be a multiplicative Vilenkin system generated by integers \(p_n\geq 2\) (also called a Price system). A set \(E\subseteq[0,1)\) is called a \(U\)-set for this system if \(\sum^\infty_{k=0} a_k\psi_k\equiv 0\) off \(E\) implies \(a_k=0\) for \(k=0,1,\dots\;\). The reviewer showed [Can. J. Math. 31, 858-866 (1979; Zbl 0415.43013)] that there exist perfect \(U\)-sets when \(p_n\equiv p\) for \(n=1,2,\dots\;\). Here the author generalizes this to any system of bounded type. Indeed, he shows perfect \(U\)-sets always exist if some subsequence \(p_{n_k}=O(k)\) as \(k\to\infty\). The author also shows that this condition is not necessary by constructing a perfect \(U\)-set for the case \(p_n=n+2\), \(n=1,2,\dots\;\). Reviewer's note: In the English translation the Russian word \textit{kontinual'noe} which should be translated as ``perfect'' has been translated as ``continual''.
scientific article; zbMATH DE number 1028803