Conditions for the uniqueness of the moment problem in the class of $q$-distributions

Let $K_q$ be the class of probability distributions on the set of non-negative integer powers of a number $q>1$ ($q$-distributions), $\mathsf P=\{p_k=P(q^k),\ k=0,1,\ldots\}$ is a distribution from the class $K_q$ which has the moments of all orders. It is shown that in order that the distribution $\mathsf P$ is uniquely determined in the class $K_q$ by the sequence of its moments provided that $p_k>0$, $k=0,1,\ldots$, it is necessary, and under the condition that
$$ \operatornamewithlimits{sup\,lim}_{k\to\infty} (p_kq^{\binom k2})^{1/k}<\infty, $$
sufficient, that
$$ \operatornamewithlimits{inf\,lim}_{k\to\infty} p_{2k}q^{\binom{2k}k} =\operatornamewithlimits{inf\,lim}_{k\to\infty} p_{2k+1}q^{\binom{2k+1}{2}}=0. $$

These results are applied in the study of the limit distribution of the number of solutions of a system of random homogeneous equations with equiprobable matrix of coefficients over a finite local ring of principle ideals.