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.