Stretching convex domains to capture many lattice points

Abstract

We consider an optimal stretching problem for strictly convex domains in ${\mathbb{R}}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant $1$. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the $(d-1)$-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes & Freitas, van den Berg, Bucur & Gittins, Ariturk & Laugesen, van den Berg & Gittins, and Gittins & Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $\#\{(i,j)\in {\mathbb{Z}}^2:i^2+j^2\le r^2\}=\pi r^2+\mathcal{O}(r^{2/3})$ result for the Gauss circle problem.

21 pages, 7 figures

References in corpus (3)

• 100 years of Weyl's law
• Maximising Neumann eigenvalues on rectangles
• Minimising Dirichlet eigenvalues on cuboids of unit measure

Cited by in corpus (5)

• Optimal stretching for lattice points and eigenvalues
• Lattice points in stretched model domains of finite type in ${\mathbb{R}}^d$
• Shifted lattices and asymptotically optimal ellipses
• Asymptotic behaviour of cuboids optimising Laplacian eigenvalues
• Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles