Harmonic Analysis and Applications

June 4-8, 2018


"Convex domains, lattice points, and Fourier analysis"

Marshall, Nicholas

We consider an optimal stretching problem for strictly convex domains in 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 Z^2 : i^2 +j^2 <= r^2 } = pi r^2 + O(r^2/3) result for the Gauss circle problem.

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