Optimality of CKP-inequality in the critical case

It is proved that the CKP inequality \[ log ⁡ N ( cov ( T ) , 2 ε ) ⪯ 1 ε ∫ ε / 2 ∞ log ⁡ N ( T , r ) d r \sqrt {\log N(\textrm {cov}(T),2\varepsilon )} \preceq \frac 1\varepsilon \int _{\varepsilon /2}^\infty \sqrt {\log N(T,r)} dr \] is optimal in the critical case log ⁡ N ( T , ε ) = O ( ε − 2 | log ⁡ ε | − 2 ) \log N(T,\varepsilon )=O(\varepsilon ^{-2}|\log \varepsilon |^{-2}) as ε → 0 + \varepsilon \to 0^+ .