Uniqueness and quasimeasures on the group of integers of a $p$-series field

Let $G$ be the group of integers of a $p$-series field and suppose that $S$ is a character series on $G$. If ${N_1}$, ${N_2}, \ldots$ is any sequence of integers and if ${S_{{p^{{N_j}}}}} \to 0$ a.e. on $G$, as $j \to \infty$, then $S$ will be the zero series provided $S$ never diverges unboundedly.