2004/07/04 by Igor Wigman, Wigman, Igor
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Quantum chaos and dynamical systems #math.NT
paper · pdf · doi:10.48550/arxiv.math/0407049
openalex publication_date 2004/07/04 · arxiv created 2005/08/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⋅ ρ). We show that if ρ shrinks slowly to zero then the distribution of the normalized counting function (N(t, ρ) - A(2tρ+ρ2))/(√(8 πβ⋅ t ρ)) is Gaussian, where A is the area of the ellipse. The case of \underlinecircular annuli is due to Hughes and Rudnick.