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The main focus of this work is the asymptotic behavior of mass-conservativehomogeneous fragmentations. Considering the logarithm of masses makes thesituation reminiscent of branching random walks. The standard approach is tostudy {\bf asymptotical} exponential rates. For fixed $v > 0$, either thenumber of fragments whose sizes at time $t$ are of order $\e^{-vt}$ isexponentially growing with rate $C(v) > 0$, i.e. the rate is effective, or theprobability of presence of such fragments is exponentially decreasing with rate$C(v) < 0$, for some concave function $C$. In a recent paper, N. Krellconsidered 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 wecontinue this analysis and focus on probabilities of presence, using the spinemethod and a suitable martingale.

Martingales and Rates of Presence in Homogeneous Fragmentations