The distribution of lattice points in elliptic annuli

Abstract

Let $N(t, ρ)$ be the number of lattice points in a thin elliptical annuli. We assume the aspect ratio $β$ of the ellipse is transcendental and Diophantine in a strong sense (this holds for {\em almost all} aspect ratios). The variance of $N(t, ρ)$ is $t(8πβ\cdot ρ)$. We show that if $ρ$ shrinks slowly to zero then the distribution of the normalized counting function $\frac{N(t, ρ) - A(2tρ+ρ^2)}{\sqrt{8 πβ\cdot t ρ}}$ is Gaussian, where A is the area of the ellipse. The case of \underline{circular} annuli is due to Hughes and Rudnick.