Lattice points in stretched model domains of finite type in ${\mathbb{R}}^d$

Abstract

We study an optimal stretching problem for certain convex domain in ${\mathbb{R}}^d$ ($d\ge 3$) whose boundary has points of vanishing Gaussian curvature. We prove that the optimal domain which contains the most positive (or least nonnegative) lattice points is asymptotically balanced. This type of problem has its origin in the "eigenvalue optimization among rectangles" problem in spectral geometry. Our proof relies on two-term bounds for lattice counting for general convex domains in ${\mathbb{R}}^d$ and an explicit estimate of the Fourier transform of the characteristic function associated with the specific domain under consideration.

arXiv comment13 pages

References in corpus (5)

• Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic
• Maximising Neumann eigenvalues on rectangles
• Minimising Dirichlet eigenvalues on cuboids of unit measure
• Stretching convex domains to capture many lattice points
• Shifted lattices and asymptotically optimal ellipses