Two-Sector Silver: The Maximizers of min{c(θ), c(θ/2)} and a Parity Criterion

For an irrational θ let c(θ) = liminf q‖qθ‖ and m(θ) = min{c(θ), c(θ/2)}, so that m(θ) = 1/λ₂(θ/2) for the 2-Lagrange approximability λ₂ of O'Dorney. The optimal value sup m = 1/√8 and the maximizing class (θ and θ/2 both GL₂(ℤ)-equivalent to √2) are known: min L₂ = 2√2 (Dong, Dupont, O'Dorney and Waitkus), and the Markoff version of the problem is A. Schmidt's 2-minimum spectrum, whose least element is √8. The main result is a parity criterion for the maximizers: if θ ~ √2 has primitive minimal polynomial Aθ² + Bθ + C, then m(θ) = 1/√8 if and only if C is even, if and only if the numerators of the continued-fraction convergents of θ are eventually all odd. For example, 2 − √2 is a maximizer, while √2 − 1, in the same GL₂(ℤ)-class, is not (it has m = 1/√32). A short self-contained proof of the optimum is given, using only the first two Markoff values and the fact that 2 is inert in ℚ(√5). The correspondence m(η) = c(H₄; η/√2)/√2, where c(H₄; ·) is the approximation constant of the Hecke group H₄, is written out; under it, the optimum corresponds to the H₄ Hurwitz constant 2. The question arose from a Hopf-type flow whose central ℤ₂ sector degenerates at a rational rotation number p/q exactly when p is even.