Stable lattices and the diagonal group

TL;DR

This paper proves that diagonal group orbits in lattice space accumulate on stable lattices, confirming a conjecture about the Mordell constant and providing new proofs for Minkowski's conjecture in low dimensions.

Contribution

It establishes the accumulation of diagonal group orbits on stable lattices and links Minkowski's conjecture to a geometric problem, offering new proofs in dimensions up to 7.

Findings

Diagonal group orbits accumulate on stable lattices.

Confirmed a conjecture of Ramharter regarding the Mordell constant.

Provided two new proofs of Minkowski's conjecture in dimensions ≤ 7.

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.