Peter Humphries

Small Scale Equidistribution of Lattice Points on the Sphere

Consider the projection onto the unit sphere in $\mathbb{R}^3$ of the set of lattice points $(x_1, x_2, x_3) \in \mathbb{Z}^3$ lying on the sphere of radius $\sqrt{n}$. Duke and Schulze-Pillot showed in 1990 that these points equidistribute on the sphere as $n \to \infty$. We study a small scale refinement of this theorem, where one asks whether these points equidistribute in subsets of the sphere whose surface area shrinks as $n$ grows. A particular case of this is a conjecture of Linnik, which states that for all $\delta >0$, the equation $x_1^2 + x_2^2 + x_3^2 = n$ has a solution with $|x_3|< n^{\delta}$ for all sufficiently large $n$. We make nontrivial progress towards this, as well as proving an averaged form of this conjecture. This is joint work with Maksym Radziwiłł.