Sharpness of the Shifted-Spectrum SOS Bound for Lyapunov Dimension

Establish whether the sum-of-squares relaxation based on the shifted spectrum approach for directly bounding Lyapunov dimension (Proposition \cref{thm:shifted dimension sos}) is sharp in general; that is, prove or refute that the resulting upper bounds converge to the exact global Lyapunov dimension as polynomial degrees are increased (or under suitable compactness and smoothness assumptions).

Background

The paper presents two SOS-based approaches to bound Lyapunov dimension: a sphere projection method, for which sharpness is established under Archimedean and compactness conditions, and a shifted spectrum method.

While sharpness (convergence to exact values as polynomial degrees increase) is proven for most formulations, the authors explicitly note that they do not provide a proof of sharpness for the shifted spectrum SOS formulation that directly bounds Lyapunov dimension, leaving its general sharpness unresolved.