Limit theorems for the number of solutions of a system of random linear equations belonging to a given set

We investigate the asymptotic behaviour of the distribution of the number $\xi(B)$ of the solutions of a system of homogeneous random linear equations $Ax=0$ (the $T\times n$ matrix $A$ is composed of independent random variables $a_{i,j}$ uniformly distributed on a set of elements of a finite field $K$) which belong to some given set $B$ of non-zero $n$-dimensional vectors over the field $K$. We consider the case where, under a concordant growth of the parameters $n,T\to\infty$ and variations of the sets $B_1,\dots,B_s$ such that the mean values converge to finite limits, the limit distribution of the vector $(\xi(B_1),\dots,\xi(B_s))$ is an $s$-dimensional compound Poisson distribution. We give sufficient conditions for this convergence and find parameters of the limit distribution. We consider in detail the special case where $B_k$ is the set of vectors which do not contain a certain element $k\in K$.