## Abstract

Following the methods of G. Gát, in this work we prove the a.e convergence of the subsequence
$\begin{array}{}({\sigma}_{\frac{{m}_{n}}{2}{M}_{n}}f{)}_{n}\end{array}$,
for every integrable function *f* on unbounded Vilenkin groups, such that the sequence (*m _{n}*)

_{n}contains infinitely many even terms satisfying the estimate $\begin{array}{}\frac{\mathrm{ln}{m}_{k-1}\mathrm{ln}{m}_{k}}{{m}_{k}}=O(1).\end{array}$

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