Semisimple Random Walks on the Torus

The candidate’s target statement exactly matches Theorem 1.1 of He–de Saxcé, including the role of E, W0=(a0γE)⊥, the Lyapunov quasi-distance d̃, and the polynomial bounds on q and v. The paper proves it by: (i) Fourier decay and a quantitative Wiener/granulation step (Proposition 5.1 and Corollary 5.5), and (ii) bootstrapping concentration plus a diophantine property and a Margulis-type instability inequality to force proximity to ZQ (Theorem 6.1, Lemmas 6.5–6.6). The model’s four-step outline (coset reduction; granulation near W0; exponential instability; parameter balancing) mirrors the paper’s proof essentially line-by-line, differing only in presentation (e.g., the paper phrases instability via a drift/“Margulis inequality” rather than an explicit r^α + e^{-cn} bound). See the paper’s statement and proof scaffolding in Theorem 1.1, Proposition 5.1/Corollary 5.5, and Theorem 6.1 with Lemmas 6.5–6.6 .