On small masses in self-similar fragmentations

We consider a self-similar fragmentation process which preserves the total mass. We are interested in the asymptotic behavior as ε→0+ of N(ε,t)=Card{i:Xi(t)>ε}, the number of fragments with size greater than ε at some fixed time t>0. Under a certain condition of regular variation type on the so-called dislocation measure, we exhibit a deterministic function ϕ:]0,1[→]0,∞[ such that the limit of N(ε,t)/ϕ (ε) exists and is non-degenerate. In general the limit is random, but may be deterministic when a certain relation between the index of self-similarity and the dislocation measure holds. We also present a similar result for the total mass of fragments less than ε.