### Spacing statistics for lattice points on circles

Heilbronn Number Theory Seminar

1st February 2023, 4:00 pm – 5:00 pm

Fry Building, 2.04

In this talk I will describe the distribution of lattice points lying on circles. A striking result of Kátai and Környei shows that along a density one subsequence of admissible radii the angles of lattice points lying on circles are uniformly distributed in the limit as the radius tends to infinity. Their result goes further, proving that uniform distribution persists even at very small scales, meaning that the angles are uniformly distributed within quickly shrinking arcs. A more refined problem is to understand how the lattice points are spaced together at the local scale, e.g. given a circle containing N lattice points determine the number of gaps between consecutive angles of size less than 1/N. I will discuss some recent joint work with Pär Kurlberg in which we compute the nearest neighbor spacing of the angles along a density one subsequence of admissible radii.

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