  The main focus of this work is the asymptotic behavior of mass-conservative
homogeneous fragmentations. Considering the logarithm of masses makes the
situation reminiscent of branching random walks. The standard approach is to
study {\bf asymptotical} exponential rates. For fixed $v > 0$, either the
number of fragments whose sizes at time $t$ are of order $\e^{-vt}$ is
exponentially growing with rate $C(v) > 0$, i.e. the rate is effective, or the
probability of presence of such fragments is exponentially decreasing with rate
$C(v) < 0$, for some concave function $C$. In a recent paper, N. Krell
considered fragments whose sizes decrease at {\bf exact} exponential rates,
i.e. whose sizes are confined to be of order $\e^{-vs}$ for every $s \leq t$.
In that setting, she characterized the effective rates. In the present paper we
continue this analysis and focus on probabilities of presence, using the spine
method and a suitable martingale.
