**“Domains capturing many lattice points: from triangles to convex domains”**

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*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.*