Stable lattices and the diagonal group

Abstract

Inspired by work of McMullen, we show that any orbit of the diagonal group in the space of lattices accumulates on the set of stable lattices. As consequences, we settle a conjecture of Ramharter concerning the asymptotic behaviour of the Mordell constant, and reduce Minkowski's conjecture on products of linear forms to a geometric question, yielding two new proofs of the conjecture in dimensions up to 7.

This paper was part of the paper "On stable lattices and the diagonal group" (arXiv:1309.4025) which was split into three shorter papers (now all on arXiv) following the referee's request