Lattice point problem and the question of estimation and detection of smooth functions of many variables

We consider the problem of asymptotics of $N_d(m)$, where $N_d(m)$ is the number of integer lattice points in the $d$-dimensional ball of radius $m$ (in $l_1$ and $l_2$-norms) for $d\to\infty$, $m\to\infty$. We show that this asymptotics differs from the asymptotic volume of $d$-dimensional ball of radius $m$ when the rate of convergence of $d$ to infinity is sufficiently high in comparison with that of $m$.