SHARPENING VAHLEN’S RESULT IN DIOPHANTINE APPROXIMATION

The paper rigorously sharpens Vahlen’s bound by working in Nakada’s natural extension and mapping (t_n, v_n) to Jager pairs (Θ_{n-1}, Θ_n), then optimizing over explicit quadrilateral regions Ψ(V_a) ∩ Ψ(H_b). This yields: (i) if m = M ≥ 2, min{Θ_{n-1}, Θ_n} ≤ m/(m^2+1) and max{Θ_{n-1}, Θ_n} ≥ (m+1)/((m+1)^2+1); (ii) if 1 ≤ m < M, min{Θ_{n-1}, Θ_n} ≤ (m+1)/((m+1)M+1) and max{Θ_{n-1}, Θ_n} ≥ M/((m+1)M+1) (Theorem 2.1). In the difficult case (a_n, a_{n+1}) = (1,1), the paper gives sharp m, M dependent bounds (Theorem 3.2). These statements and their geometric proofs are clearly laid out via Θ_n = t_n/(1+t_nv_n) and its companion formula (Eqs. (5)–(6)) and the Ψ-geometry (figures and vertex computations) . The candidate solution uses a Perron-type parameterization Θ_n = 1/(α_{n+1}+β_n) with α_{n+1} = a_{n+1}+ξ (0<ξ<1) and β_n = 1/(a_n+η) (0<η<1). Its Case B (a_n=a_{n+1}=1) matches the paper’s Theorem 3.2 exactly. However, in Case A it (1) invokes an incorrect inequality D1+D2 ≥ a_{n+1}+a_n + 1/a_n + 1/a_{n+1} (the direction reverses; 1/(a+η) ≤ 1/a), and uses it to claim min{Θ_{n-1},Θ_n} ≤ m/(m^2+1); and (2) fails to recover the paper’s sharper M-dependent bounds in Theorem 2.1(ii), giving only a weaker max{Θ_{n-1},Θ_n} ≥ (m+1)/((m+1)^2+1) (the paper achieves ≥ M/((m+1)M+1) when m<M) . The paper’s arguments are coherent and complete; the model’s Case A contains a critical inequality error and misses the M-dependence.