The distribution of lattice points in elliptic annuli

Abstract

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