The Exact $\epsilon$-Hypercyclicity Threshold

In this paper we give an affirmative answer to the problem proposed by Bayart in [J. Math. Anal. Appl. $\mathbf{\text{529}}$ (2024), 127278]: given $\epsilon \in (0,1)$, there exists an operator which is $\delta$-hypercyclic if and only if $\delta \in [\epsilon ,1)$?