Determine the asymptotic behavior of the hierarchy truncation error

Determine the true asymptotic behavior of the truncation error \(\Lambda-\Lambda_M\) for the finite-arc Reuleaux hierarchy, beyond the established \(O(M^{-2})\) upper bound.

Background

The paper constructs a monotone finite-arc hierarchy $\Lambda_M$ converging to Lebesgue’s universal covering constant $\Lambda$, and proves the quantitative estimate $0\leq \Lambda-\Lambda_M\leq C M^{-2}$. The paper also establishes that the corresponding Hausdorff approximation scale is $\Theta(M^{-2})$, but this does not determine the actual asymptotic behavior of the covering-area error.

The unresolved issue is whether the covering-area functional smooths or otherwise improves upon the geometric $M^{-2}$ approximation scale. Determining the precise asymptotics of $\Lambda-\Lambda_M$ could clarify the efficiency of finite-arc truncations and guide the design of sharper certified lower-bound computations.