Shifted lattices and asymptotically optimal ellipses

Abstract

Translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. Take a decreasing concave (or convex) curve in the first quadrant and construct a family of curves by rescaling in the coordinate directions while preserving area. Consider the curve in the family that encloses the greatest number of the shifted lattice points: we seek to identify the limiting shape of this maximizing curve as the area is scaled up towards infinity. The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead we show it degenerates. Our results handle the $p$-circle $x^p+y^p=1$ when $p>1$ (concave) and also when $0<p<1$ (convex). Rescaling the $p$-circle generates the family of $p$-ellipses, and so in particular we identify the asymptotically optimal $p$-ellipses associated with shifted integer lattices. The circular case $p=2$ with shift $-1/2$ corresponds to minimizing high eigenvalues in a symmetry class for the Laplacian on rectangles, while the straight line case ($p=1$) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.