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

V. A. Kopyttsev

Diskretnaya Matematika, Tome 19 (2007) no. 4, pp. 3-22

Résumé

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.

Bibliographie

[1] Kopyttsev V. A., “O chisle reshenii sistem lineinykh bulevykh uravnenii v mnozhestve vektorov, obladayuschikh zadannym chislom edinits”, Diskretnaya matematika, 14:4 (2002), 87–109 | MR | Zbl

[2] Kopyttsev V. A., “O chisle reshenii sistemy sluchainykh lineinykh uravnenii v mnozhestve vektorov spetsialnogo vida”, Diskretnaya matematika, 18:1 (2006), 40–62 | MR | Zbl

[3] Mikhailov V. G., “O chisle reshenii, blizkikh k zadannomu vektoru”, Obozrenie prikladnoi i promyshlennoi matematiki, 12:2 (2005), 435–436

[4] Mikhailov V. G., “Skhodimost k protsessu s nezavisimymi prirascheniyami v skheme narastayuschikh summ zavisimykh sluchainykh velichin”, Matem. sb., 94 (1974), 283–299