Normal and Lognormal Asymptotics for Lattice-Point Counts in Random Translates of High-Dimensional Balls

Let the center of a $d$-dimensional Euclidean ball be uniformly distributed modulo ${\mathbb{Z}}^d$. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is $4{\pi }^2{R}_d^2\mathrm{/}(d+2)=\log d$. In a logarithmic window around this scale, the logarithm of the lattice-point count divided by the volume satisfies a central limit theorem. At a fixed offset from the critical scale this yields a genuine lognormal limit. Above the critical scale, the count-to-volume ratio converges to $1$ in measure; its normalized error has a standard normal limit, and its variance satisfies an asymptotic formula.