Density of shapes of periodic tori in the cubic case

2025/02/18 by Dang, Nguyen-Thi, Gargava, Nihar, Li, Jialun · 2 citations
#37A44 #37PXX #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.12754

Abstract

Consider the compact orbits of the ℝ² action of the diagonal group on SL(3,ℝ)/SL(3,ℤ), the so-called periodic tori. For any periodic torus, the set of periods of the orbit forms a lattice in ℝ². Such a lattice, re-scaled to covolume one, gives a shape point in SL(2,ℝ)/SL(2,ℤ). We prove that the shapes of all periodic tori are dense in SL(2,ℝ)/SL(2,ℤ). This implies the density of shapes of the unit groups of totally real cubic orders.

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• Classification of totally real number fields via their zeta function, regulator, and log unit lattice
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