Effective joint equidistribution of primitive rational points on expanding horospheres

Abstract

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least three dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems – an approach which does not lend itself to effective bounds. We implement a strategy based on spectral theory, Fourier analysis and Weil’s bound for Kloosterman sums in order to quantify the rate of equidistribution for a specific horospherical subgroup in any dimension. We apply our result to provide a rate of convergence to the limiting distribution for the appropriately rescaled diameters of random circulant graphs.