Density of shapes of periodic tori in the cubic case
Nguyen-Thi Dang, Nihar Gargava, Jialun Li
TL;DR
The paper proves that the shapes of periodic tori arising from the diagonal $A$-action on $\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$ are dense in $\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})$, linking dynamics to number-theoretic shapes of unit groups of totally real cubic orders. It builds a concrete path via a family of simplest cubic fields $f(X)=X(X-a_1)(X-a_2)-1$, showing the unit group is generated by explicit elements and controlling the shape with log-lengths of units; suborder inclusions yield a $3\times 2$ matrix congruence governing new shapes. The core argument combines lower bounds for regulators (Cusick), Smith normal form analysis of local/global congruences, and the banana trick to deduce density, with a density result established first for a $(2,3,5)$-based family and then extended to all suborders of the constructed family. This provides a bridge between volume- and systole-based counting in higher-rank homogeneous dynamics and yields dense coverage of unit-shape loci in the moduli of flat tori, with potential implications for equidistribution-type questions in arithmetic dynamics.
Abstract
Consider the compact orbits of the $\mathbb{R}^2$ action of the diagonal group on $\operatorname{SL}(3,\mathbb{R})/\operatorname{SL}(3,\mathbb{Z})$, the so-called periodic tori. For any periodic torus, the set of periods of the orbit forms a lattice in $\mathbb{R}^2$. Such a lattice, re-scaled to covolume one, gives a shape point in $\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$. We prove that the shapes of all periodic tori are dense in $\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$. This implies the density of shapes of the unit groups of totally real cubic orders.