Let ω=(ωn)n[ges ]1 be a log concave sequence such that lim infn→+∞ωn/nc>0 for some c>0 and ((log ωn)/nα)n[ges ]1 is nonincreasing for some α<1/2. We show that, if T is a contraction on the Hilbert space with spectrum a Carleson set, and if ∥T−n∥=O(ωn) as n tends to +∞ with [sum ]n[ges ]11/(n log ωn)=+∞, then T is unitary. On the other hand, if [sum ]n[ges ]11/(n log ωn)<+∞, then there exists a (non-unitary) contraction T on the Hilbert space such that the spectrum of T is a Carleson set, ∥T−n∥=O(ωn) as n tends to +∞, and lim supn→+∞∥T−n∥=+∞.