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In this talk, we will introduce a new notion of width for convex bodies. Given a fixed convex body $K$ and a linear functional, we define the arithmetic width of $K$ with respect to this direction to be the size of the image of the lattice points of $K$ under this linear functional. This notion has natural connections to existing notions of width, namely the lattice width, but accounts for gaps in the polytope in integer directions. We will introduce a set that we call the arithmetic range of $K$ and discuss recent results showing that this set forms an almost arithmetic progression for sufficiently large dilations of $K$. Lastly, we discuss our on-going work showing that the arithmetic width of $nP$ is a quasipolynomial in $n$ when $P$ is a rational polytope. This is joint work with Jesús A. De Loera and Christopher O’Nei.

Free pizzas as always:)