On the uniqueness of the moment problem in the class of $q$-distributions

We introduce the notion of $\overline{q}=(q_1,\ldots,q_t)$-binomial invariants of a random vector $\overline{\xi}=(\xi_1,\ldots,\xi_t)$ distributed on the set of vectors whose $i$th coordinate is a non-negative integer power of the number $q_i>1$, $i=1,\ldots,t$. We find relations between $\overline{q}$-binomial invariants and mixed moments, give conditions under which the distribution of the vector $\overline{\xi}$ is uniquely determined by the sequence of its $\overline{q}$-binomial invariants, and present expressions of probabilities
$$ \mathsf P(\xi_1=q_1^{r_1},\ldots,\xi_t=q_t^{r_t}),\qquad r_1,\ldots,r_t=0,1,\ldots, $$
in terms of the corresponding $\overline{q}$-binomial invariants and estimates of these probabilities. We prove a theorem on convergence of a sequence of distributions of such random vectors under the condition that the corresponding $\overline{q}$-binomial invariants converge.
The results obtained are used in the study of the limit distribution of the number of solutions of a class of systems of linear equations over a finite field.