Lattice Point Problem and Questions of Estimation and Detection of Smooth Multivariate Functions

Abstract

Let N_d(m) be the number of points of the integer lattice that belong to a d-dimensional ball of radius m (in the l₁- and l₂-norms). The aim of the paper is to study the asymptotic behavior of N_d(m) as d → ∞, m → ∞. It is shown that if d tends to infinity much faster than m, then the asymptotic is different from the asymptotic volume of a d-dimensional ball of radius m. Bibliography: 6 titles.