ABOUT CHARACTERS ON VILENKIN GROUPS

Abstract

In this paper we proved that Lemma [7, p.727] is not true and that the following Theorem holds: Let \(G\) be Vilenkin group. Let \(k \in [m_n,m_{n+1})\) be natural number given by
\[
k=\sum_{j=0}^{n}k_jm_j,\qquad 1\leq k_n<p_{n+1};\quad 0\leq k_j<p_{j+1},
\]
\(j\in\{0,1,2,\ldots,n-1\}\) and let \(\bar{k}\) be natural number such that \((\forall x\in G)\)
\[
\chi_k(x)\cdot\chi_{\bar{k}}(x)=1.
\]
In this case \(\bar{k}=m_{n+1}-k+m_n-1\). Additionally, we have proved lemma which can be useful independently from our theorem.