Example of a null-series with respect to periodic multiplicative systems (Q1896975)

The authors consider the multiplicative Vilenkin orthonormal system \(\{\varphi_n(t): n= 0,1,\dots\}\) defined on a zero-dimensional compact Abelian group satisfying the second axiom of countability, and determined by the sequence \(\{p_n\}\) of primes. They construct an \(M\)-set of measure zero with respect to the system \(\{\varphi_n(t)\}\), without imposing any restriction on \(\{p_n\}\). As it is known, a set \(E\subset G\) is called an \(M\)-set with respect to \(\{\varphi_n(t)\}\) if there exists a series \(\sum a_n\varphi_n(t)\) with \(a_n\neq 0\) for at least one \(n\), which converges to zero everywhere outside \(E\). The construction follows in great lines that of the corresponding \(M\)-set for the Walsh system (when \(p_1= p_2=\cdots= 2\)).