Equidistribution of rational points on shrinking sets of S3: A juxtaposition of an automorphic and a circle method approach

Raphael Steiner (University of Bristol)

Frank Adams 1,

It is a classical theorem in the theory of modular forms that the points \(\boldsymbol{x}/\sqrt{N}\), where \(\boldsymbol{x} \in \mathbb{Z}^n\) runs over all the solutions to \(\sum_{i=1}^n x_i^2=N\), equidistribute on \(S^{n-1}\) for \(n \ge 4\) as \(N\) (odd) tends to infinity. The rate of equidistribution poses however a more challenging problem. Due to its Diophantine nature the points inherit a repulsion property, which opposes equidistribution on small sets. Sarnak conjectures that this Diophantine repulsion is the only obstruction to the rate of equidistribution. We shall compare two approaches to this problem: an automorphic approach, where we make use of spherical functions, and a circle method approach, given in the form of the smooth delta symbol circle method.

 

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