Effective Joint Equidistribution of Subspaces of Bounded Height in Rigid Adelic Spaces

Abstract

We prove an effective joint equidistribution theorem for $d$-dimensional $K$-subspaces, $1\le d<n$, of a fixed $n$-dimensional rigid adelic space over a number field $K$. The parameters are the height, the image in the archimedean Grassmannian, and the isometry classes of the subspace and quotient after each is rescaled to height $1$. For bounded Lipschitz test functions, the error term is power-saving with an explicit exponent. As an application, our theorem refines Thunder's asymptotic formula for subspaces of bounded height. We also obtain a power-saving asymptotic for subspaces containing $d$ linearly independent projective points of bounded height, with a leading constant expressed as a moment of the last successive minimum.

arXiv commentThe v2 paper upgrades the oringinal copy to an effective equidistribution theorem. The old version is conceptually easier to read though. The new version has stronger theorem but very technical