Exactness of weighted exponential systems with a defect

Let $w\text{\hspace{0.17em}⁣}:(0,1)\to {\mathbb{R}}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i{\lambda }_{n}t}{\}}_{n\in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A\subset \mathbb{Z}$ the system $\{w(t)r_n(t){\}}_{n\in \mathbb{Z}\setminus A}$ 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 $\{w(t)e^{2\pi int}{\}}_{n\in \mathbb{Z}\setminus A}$ to be complete and minimal in $L^2(0,1)$ for an arbitrary weight $w$ and a finite defect set $A\subset \mathbb{Z}$.