The distribution of lattice points in elliptic annuli

Let $N(t, \rho)$ be the number of lattice points in a thin elliptical annuli. We assume the aspect ratio $\beta$ of the ellipse is transcendental and Diophantine in a strong sense (this holds for {\em almost all} aspect ratios). The variance of $N(t, \rho)$ is $t(8\pi \beta \cdot \rho)$. We show that if $\rho$ shrinks slowly to zero then the distribution of the normalized counting function $\frac{N(t, \rho) - A(2t\rho+\rho^2)}{\sqrt{8 \pi \beta \cdot t \rho}}$ is Gaussian, where A is the area of the ellipse. The case of \underline{circular} annuli is due to Hughes and Rudnick

Optimal lattice configurations for interacting spatially extended particles

We investigate lattice energies for radially symmetric, spatially extended particles interacting via a radial potential and arranged on the sites of a two-dimensional Bravais lattice. We show the global minimality of the triangular lattice among Bravais lattices of fixed density in two cases: In the first case, the distribution of mass is sufficiently concentrated around the lattice points, and the mass concentration depends on the density we have fixed. In the second case, both interacting potential and density of the distribution of mass are described by completely monotone functions in which case the optimality holds at any fixed density.Comment: 17 pages. 1 figure. To appear in Letters in Mathematical Physic