Super-polynomial decay with infinite defect sets

Determine whether, for an arbitrary infinite defect set and a weight with super-polynomial decay at zero, the weighted system of dual exponentials obtained by multiplying the biorthogonal system of a Schauder basis in $L^2(0,1)$ by the weight is always complete and never minimal.

Background

The paper studies weighted exponential systems formed from the biorthogonal system of an exact exponential system or Schauder basis after removing a defect set of frequencies. For finite defect sets, the authors note that a weight with super-polynomial decay at zero forces the resulting weighted system to be complete but not minimal. The unresolved issue is whether an analogous conclusion holds when the defect set is infinite.

The paper resolves this question only for infinite defect sets whose corresponding spectral set is two-sided lacunary, proving completeness and non-minimality under a quantitative decay condition. Thus, the broader case of arbitrary infinite defect sets and super-polynomially decaying weights remains open.