Singular vectors on manifolds over number fields

The paper proves Theorem 1.4—under Besicovitch/Federer and good/nonplanar hypotheses for a product map f, the pushforward measure assigns zero mass to singular vectors—via an S-arithmetic Dani correspondence (Theorem 3.2) and a quantitative nondivergence (QND) estimate specialized to OK-modules (Theorems 4.2–4.3), culminating in Proposition 4.6 and a local-to-global argument. This chain is explicit and coherent in the manuscript, with the theorem statement and assumptions spelled out and the proof relying on a sequence of times ti (not uniform in t) to obtain the needed lower bounds uniformly over all primitive submodules Δ, then invoking Dani to pass from small δ at some time to divergence and hence singularity; the set of singular points in each ball has measure zero, and coverings finish the proof . In contrast, the candidate solution asserts a uniform-in-t QND bound on μ({x: g_t u_{f(x)}Γ ∉ K_ε}) using a single ρ>0 chosen “uniformly over Δ,” and then applies time-averaging/Fubini to preclude divergence. That step is not justified: the required lower bound sup_{x∈B} cov(g_t u_{f(x)}Δ) ≥ ρ^{rk(Δ)} for all Δ and all t cannot be obtained by appealing to “finitely many wedge patterns,” since Δ ranges over infinitely many primitive OK-submodules and the relevant coefficients c(w) vary; the paper handles this precisely by using a sequence of times ti to achieve a uniform c>0 across all Δ (Proposition 4.6) rather than a bound holding for every t . Thus, while the candidate’s setup (Dani correspondence, wedge/“good” verification) matches the paper in spirit, the key uniformity claim is flawed; without it, the time-averaging argument does not go through, whereas the paper’s discrete-time QND approach does.