Shifted lattices and asymptotically optimal ellipses

Translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions, and consider a decreasing concave (or convex) curve in the first quadrant. Construct a family of curves by rescaling in the coordinate directions while preserving area, and identify the curve in the family that encloses the greatest number of the shifted lattice points. We find the limiting shape of this maximizing curve as the area is scaled up towards infinity. The limiting shape depends explicitly on the lattice shift, except that when the shift is too negative, the maximizing curve fails to converge and instead degenerates. Our results handle the p-circle x ^p+ y ^p= 1 when p> 1 (concave) and also when 0 < p< 1 (convex). 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.