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 behavior 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.
Cite this article
Uri Shapira, Barak Weiss, Stable lattices and the diagonal group. J. Eur. Math. Soc. 18 (2016), no. 8, pp. 1753–1767DOI 10.4171/JEMS/628