Lattice point counting statistics for 3-dimensional shrinking Cygan-Korányi spherical shells

Abstract

Let $E(x;ω)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Korányi spherical shell of inner radius $x$ and gap width $ω(x)>0$. Assuming that $ω(x)\to0$ as $x\to\infty$, and that $ω$ satisfies suitable regularity conditions, we prove that $E(x;ω)$, properly normalized, has a limiting distribution. Moreover, we show that the corresponding distribution is moment-determinate, and we give a closed form expression for its moments. As a corollary, we deduce that the limiting distribution is the standard Gaussian measure whenever $ω$ is slowly varying. We also construct gap width functions $ω$, whose corresponding error term has a limiting distribution that is absolutely continuous with a non-Gaussian density.