Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle

The paper’s Theorem 1.3 states explicitly that the set of weighted uniform inhomogeneous Dirichlet pairs D̂_{n,m}[g;α,β] has Lebesgue measure zero if the series ∑_{l≥1} (1/l)·(∏_j β_j(l))·(∏_i α_i(g(l))) diverges, and full measure if it converges . This agrees with the unweighted statement (Theorem A / Proposition 1.7), which in the weights setting reads: the set of (g; ρ,σ)-Dirichlet pairs has zero measure if ∑_{l≥1} 1/(l^2 g(l)) diverges, and full measure if it converges . The candidate’s counterexample relies on a claim that uniform inhomogeneous Dirichlet (e.g., g(T)=1/T in dimension 1) holds for every pair (Θ,η), which the paper’s introduction explicitly refutes: not all pairs are f1-Dirichlet, and in general ||Θq−η|| can be bounded away from zero . Moreover, the proof strategy in the paper uses transference between uniform inhomogeneous pairs and homogeneous approximability (Theorems 1.4 and 1.5) to establish the zero–one law with the “divergence ⇒ null” direction for pairs; see the proof of Theorem 1.3 . Hence the paper’s direction is correct, and the model’s reversal and its “counterexample” are invalid.