Badly Approximable Grids and k-Divergent Lattices

The paper defines DVF by the exact equality VF(y) = [0, ∞), which cannot hold for any grid since VF(y) is countable, and it also relies on an Inheritance Lemma whose proof only yields inclusion of closures, not of the sets themselves. It further asserts that F is h_t-invariant, which conflicts with its own definition of h_t unless m = n. At the same time, the paper’s “moreover” statement—that for µ-a.e. y the forward orbit-closure contains a coset of a (d−1)-dimensional subtorus—appears coherent within their measure-pushing framework and would imply the correct dense-values conclusion under the natural correction DVF ≡ cl(VF(y)) = [0, ∞). The candidate model correctly flags the countability obstruction and proposes the standard fix, and sketches how the ‘moreover’ conclusion yields dense values; however, parts of the model’s argument (e.g., continuity of F on a set of grids) are imprecise and need repair. Net: both sides need substantial corrections/clarifications. Key places in the paper showing issues include Definition 2.5 (equality instead of density), Lemma 2.6 (set vs. closure), the statement “F is h_t-invariant” with h_t as in (1.6), and the use of Proposition 2.7 to deduce DVF from a coset in the orbit-closure .