Twisted Diophantine approximation on manifolds

The paper’s Theorem 2 states exactly the α–twisted Khintchine-type for convergence under the threshold ω(α^T) < (n/m − n/(2m^2(m−1)))^{-1} and proves that for any doubling ψ with ∑ q^{n−1}ψ(q)^m < ∞, the set Tψ(α) has zero measure on any nondegenerate curve and on any nondegenerate analytic manifold, i.e., the manifold is of α–twisted Khintchine type for convergence over doubling ψ (Theorem 2 and (1.11) in the paper) . The proof is via dyadic reduction to T̃ψ(α), linearization of the curve, and a geometry-of-numbers analysis of intersections with the lattice ΛQ = gQuα Zm+n (Notation 2.2; Lemma 3.1; Lemma 3.3) . The “bad” set of parameters is controlled using a dual-lattice condition verified from the Diophantine assumption on α^T with a carefully chosen ∆(Q), yielding a summable bound on |SQ| and completion by Borel–Cantelli (Lemmas 3.6 and 3.7; the crucial choice of γ > ω(α^T) and equation (4.3); the summation/BC step in §4.1) . By contrast, the model’s (candidate) proof hinges on two critical points which are not justified and, in fact, lead to an incorrect error exponent: (i) it assumes a uniform polynomial Fourier-decay bound |μ̂(k)| ≲ ∥k∥^{-1/(m−1)} for pushforward (surface) measures on arbitrary nondegenerate analytic manifolds, purportedly from the Kleinbock–Margulis “good functions” theory; that theory controls small-value sets of scalar functions, not oscillatory integral decay, and does not furnish the asserted stationary-phase-type uniform bound; and (ii) its L^2 error computation misestimates the k-sum, yielding r-exponents with the wrong sign. A correct dimensional counting gives the first Cauchy–Schwarz factor scaling like r^{m/2 + 1/(m−1)} rather than the model’s r^{m − 1/2 − 1/(m−1)}, so the combined error behaves like Q^{n−1} r^{1/(m−1) − ω}, which is not summable for typical ω ≥ n/m. The model’s later optimization and derived threshold therefore lack support. The paper’s argument is complete and coherent; the candidate’s derivation depends on an unjustified Fourier-decay claim and a sign error in the exponent arithmetic.