Hereditary completeness for systems of exponentials in weighted $L^2$-spaces

We prove that for every $0 < α< 1$ there exists $Λ_α \subset \mathbb{R}$ such that the system $\{e^{iλt}\}_{λ\in Λ_α}$ is complete and minimal, but not hereditarily complete in the weighted space $L^2(w)$ for any weight $w$ with $w(t) \asymp (π-|t|)^α$ on $(-π,π)$.