On an estimate of the concentration function for the sum of identically distributed two-dimensional independent lattice random vectors

The following theorem is proved. If $\xi_1,\xi_2,\dots$ is a sequence of non-degenerate identically distributed independent random variables with values in $Z^2$, then
$$ \sup_{m\in Z^2}\mathbf P(\xi_1+\dots+\xi_n=m)\le Cn^{-1}\Delta^{-1/2}, $$
where $C$ is an absolute constant, $\Delta=(P_L-P_0)(1-P_L)$,
$$ P_0=\max_{m\in Z^2}\mathbf P\{\xi=x\},\qquad P_L=\max_H\mathbf P\{\xi\in H\}, $$
$H$ is a set of points belonging to some straight line.