Density of shapes of periodic tori in the cubic case

The uploaded preprint proves density of shapes for periodic tori in the Weyl-chamber space M\SL(3,R)/SL(3,Z) (Theorem 1.1) and, with minor modifications, for SL(3,R)/SL(3,Z) via positive units (Theorem 5.4) using exactly the arithmetic–dynamical strategy summarized by the model: Cassels/Ankeny–Brauer–Chowla-type monogenic cubic orders plus Cusick’s regulator bound to get an initial family of shapes; passage to suborders via an explicit 3×2 matrix congruence to realize many sublattices; and a diagonal-invariance/horospherical-piece argument concluded by the Kleinbock–Margulis “banana trick” to obtain density (Proposition 5.2) . Lemma 2.6 identifies the period lattice with the logarithmic unit lattice (with signs handled via M or by restricting to totally positive units), matching the model’s arithmetic identification up to a small sign-detail . The only substantive inaccuracies in the model are (i) reversing which space is handled in Section 5.4 (the paper treats positive units/SL(3) there, not the Weyl-chamber), and (ii) attributing totally positive units to Lemma 2.6 rather than to the SL(3) setting noted in Remark 2.7 and Section 5.4; these are minor citation-level errors that do not affect the core argument or conclusion .