“Domains capturing many lattice points: from triangles to convex domains”
We discuss two problems about lattice points. (1) Motivated by problems in
mathematical physics, Antunes & Freitas proved in 2012 that among all
axis-parallel ellipses in the plane centered at the origin and having fixed
area, the one containing the most lattice points asymptotically converges
to a circle as area becomes large. We give a far-ranging generalization to convex bodies with
nonvanishing Gauss curvature. The methods are based on Fourier analysis.
(2) Secondly, in joint work with S. Steinerberger, we answer a question
posed by Laugesen & Liu about right-angled triangles in the plane capturing
many lattice points. Instead of a single optimal shape, there is a countably
infinite limit set of triangles each of which capture a maximal number of
positive lattice points for arbitrarily large areas. Moreover, this limit set is
fractal with Minkowski dimension at most 3/4. The proof involves elements
from combinatorics and dynamical systems.