S-integral quadratic forms and homogeneous dynamics

The uploaded paper explicitly proves two quantitative results: (i) an S-integral equivalence criterion with an explicit bound ||γ0||S ≤ Ad pS^{19 d^6} (||Q1||∞||Q2||∞)^{2 d^3} (Theorem 1.3), and (ii) a small-generators theorem for O(Q, ZS) with ||ξ||S ≤ Bd pS^{20 d^7} ||Q||∞^{5 d^6} (Theorem 1.5) . The proofs hinge on a precise dynamical reformulation on X_{d,S} = G_{d,S}/Γ_{d,S}, uniform effective mixing on closed H◦_S-orbits (Propositions 3.2/3.3 and Proposition 3.16), and a polynomial upper bound on vol(Y_{Q,S}) (Section 4) . By contrast, the candidate’s solution appeals to a mean-ergodic/mixing argument on Γ\G_S with height balls and Property (τ), but does not supply a correct mechanism to pass from approximate to exact ZS-equivalence on a closed HS-orbit, nor does it recover the place-by-place bounds of Theorems 5.1–5.2 that are essential in the paper’s derivation of Theorem 1.3 and in the proof of Theorem 1.5 . Several steps are sketched at a heuristic level (e.g., the definition and positive measure of the target set E in Γ\G_S, and the generation argument), leaving substantive gaps.