Singular vectors in affine subspaces and Ψ−Dirichlet numbers

The paper’s Theorem 2.4 asserts that if ψ(t) < 1/t eventually, then D(ψ) = F_q(T), and proves it via continued fractions and a best-approximation criterion (Lemma 5.1) together with the identity |⟨q_{n-1}x⟩| = 1/|q_n| for convergents. This logic correctly shows that no irrational x can be ψ-Dirichlet under ψ(t) < 1/t, while rationals are trivially included, hence D(ψ) = F_q(T) as claimed. The candidate model’s counterexample ψ(t) = 1/(2t) is flawed: it relies on an inequality chain in the discrete non-archimedean norm that reverses the direction of comparison and ignores the worst-case Q arbitrarily close to |q_{n+1}|, where the minimal achievable error is 1/|q_{n+1}| and any threshold c/Q with c < 1 necessarily fails for infinitely many large Q. Thus the proposed “sharp” constant 1/e is incorrect; the sharp threshold is 1.