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 $4π^2R_d^2/(d+2)=\log d$. In a logarithmic window around this scale, the