A multivariate Poisson theorem for the number of solutions close to given vectors of a system of random linear equations

We consider the number $(\xi(A,b\mid z)$ of solutions of a system of random linear equations $Ax=b$ over a finite field $K$ which belong to the set $X_r(z)$ of the vectors differing from a given vector $z$ in a given number $r$ of coordinates (or in at most a given number of coordinates). We give conditions under which, as the number of unknowns, the number of equations, and the number of noncoinciding coordinates tend to infinity, the limit distribution of the vector $(\xi(A,b\mid z^{(1)}),\dots,\xi(A,b\mid z^{(k)}))$ (or of the vector obtained from this vector by normalisation or by shifting some components by one) is the $k$-variate Poisson law. As corollaries we get limit distributions of the variable $(\xi(A,b\mid z^{(1)},\dots,z^{(k)}))$ equal to the number of solutions of the system belonging to the union of the sets $X_r(z^{(s)})$, $s=1,\dots,k$. This research continues a series of the author's and V. G. Mikhailov's studies.