On peak sets for Hölder classes (a counterexample to E. M. Dyn'kin's conjecture)

We construct a peak set $E\subset\mathbb T$ for the analytic Hölder class $A_\alpha$ ($0<\alpha<1$) such that $\operatorname{dist}(\cdot,E)^{-\alpha}\notin L^1(\mathbb T)$.