Michael Bersudsky

Linnik's problem and statistics of orthogonal grids of primitive integral vectors

I will discuss my joint work with Uri Shapira on a “Linnik type” equidistribution problem in a non-Euclidean setting, as well as its application to a generalisation of Uri's previous work with Menny Aka and Manfred Einsiedler on the statistics of orthogonal grids.
In overview, we consider an integral polynomial on the group of unimodular matrices SL(d,R), and we study the statistics of SL(d,Z) matrices lying on a "large" level set in relation to a certain natural map to a reference level set. Taking the quotient of a level set by ASL(d-1,Z), we get a moduli space of orthogonal grids to vectors lying on a level set of a quadratic form