THE DISTRIBUTION OF LATTICE POINTS IN ELLIPTIC ANNULI

We study the distribution of the number of lattice points lying in thin elliptical annuli. It has been conjectured by Bleher and Lebowitz that if the width of the annuli tends to zero and their area tends to infinity, then the distribution of this number, normalized to have zero mean and unit variance, is Gaussian. This has been proved by Hughes and Rudnick for circular annuli whose width shrinks to zero sufficiently slowly. We prove this conjecture for ellipses whose aspect ratio is transcendental and strongly Diophantine, also assuming the width shrinks slowly to zero.