Lattice counting for deformations of convex domains

Abstract

Let D₀ = {x ∈ Rⁿ,H₀(x) ≤ 1} be a strictly convex domain in Rⁿ with n ≤ 3 and D_u = {x ∈ Rⁿ,H_u(x) ≤ 1}, u ∈ [η, η] be a continuous one-parameter deformation of D₀ with lattice-point counting function N_u(T) := {m ∈ Zⁿ : H_u(m) ≤ T²}. The main result of this paper is an estimate for large values of T of the variation of the counting function, N_u(T), over generic volume-preserving deformations D_u.