The asymptotic distribution of the joint values of the integral lattice points for a system of a quadratic form and a linear form

Abstract: Let Q be a quadratic form and let L be a linear form on the n-dimensional real vector space. We are interested in the distribution of the image of the integral lattice under the map (Q, L). Developing the celebrated work of Eskin, Margulis, and Mozes in 1998, we provide the conditions of systems of forms which satisfy that the number of integral vectors in the ball of radius T whose joint values are contained in a given bounded set converges asymptotically to the volume of the region given by the level sets of the quadratic form and the linear form, intersecting with the ball of radius T, as T goes to infinity. This condition is introduced by Gorodnik in 2004. For this, we need to classify all intermediate subgroups between the special orthogonal group preserving Q and L and the special linear group. Among them, only two closed subgroups are of our concern. We will introduce Siegel integral formulas and equidistribution theorems for each subgroup, and show how to reach our main theorem. This is joint work with Seonhee Lim and Keivan Mallahi-Karai.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar