Exactness of weighted exponential systems with a defect

Abstract: Let $w \colon (0,1) \to \mathbb{R}<em>{+}$ be a weight. We prove that for an arbitrary Schauder basis ${e^{i λ_n t}}</em>{n \in \mathbb{Z}}$ in $L<sup>2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system ${w(t) r_n(t)}<em>{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any weight $w$ satisfying a natural decay condition, which is sharp on the exponential scale. Moreover, we establish a simple combinatorial criterion for systems of the form $${ w(t) e^{2πi n t} }</em>{n \in \mathbb{Z} \setminus A}$$ to be complete and minimal in $L<sup>2(0,1)$ for an arbitrary weight $w$ and a finite defect set $A \subset \mathbb{Z}$.