The largest fragment in self-similar fragmentation processes of positive index

We study a self-similar fragmentation process with dislocation measure $\nu$ and self-similarity index $\alpha >0$. Let $e^{-m_t}$ denote the size of the largest fragment at time $t\ge 0$. For dislocation measures satisfying a regularity condition of the form $\nu (1-s_1>\delta )={\delta }^{-\theta }\mathrm{\ell }(1\mathrm{/}\delta )$ with $\theta \in [0,1)$ and slowly varying $\mathrm{\ell }$, we prove almost sure convergence $$\underset{t\to \mathrm{\infty }}{\lim }(m_t-g(t))=0,$$ where $g(t)=(\log t-(1-\theta )\log \log t+f(t))\mathrm{/}\alpha$, and $f(t)=o(\log \log t)$ is a lower order correction that can be described explicitly in terms of $\mathrm{\ell }$ and $\theta$. Our results sharpen substantially the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t=(1+o(1))\log (t)\mathrm{/}\alpha$.