Determine the finite-parameter optimal density for normal operators in logarithmic sectors

Determine the exact optimal sampling-density coefficient for normal operators satisfying the logarithmic sector condition $|\\arg\\lambda|\\leq M(-\\log|\\lambda|)$ at each finite original frame condition-number bound $\\kappa>1$, and determine its asymptotic behavior as $\\kappa\\downarrow1$.

Background

The paper studies common weighted sampling designs for normal operator orbits whose nonzero spectral values lie in a logarithmic sector, equivalently satisfying $|\operatorname{Im}L|\leq M\operatorname{Re}L$ for $L=-\operatorname{Log}A$. For fixed $M$, Theorem \ref{nrm-sharp} determines only the leading asymptotic behavior of the optimal counting coefficient as $\kappa\to\infty$, namely $$C_M^*(\kappa)\sim [\pi(\pi-2\arctan M)]^{-1}\log\kappa$$.

The unresolved task is to obtain the exact value of $C_M^*(\kappa)$ for finite $\kappa>1$ and to characterize its behavior in the nearly tight regime $\kappa\downarrow1$. This would refine the large-condition-number result and complete the parameter dependence for normal operators in logarithmic sectors.