  We study the asymptotics of the $k$-regular self-similar fragmentation
process. For $\alpha > 0$ and an integer $k \geq 2$, this is the Markov process
$(I_t)_{t \geq 0}$ in which each $I_t$ is a union of open subsets of $[0,1)$,
and independently each subinterval of $I_t$ of size $u$ breaks into $k$ equally
sized pieces at rate $u^\alpha$. Let $k^{ - m_t}$ and $k^{ - M_t}$ be the
respective sizes of the largest and smallest fragments in $I_t$. By relating
$(I_t)_{t \geq 0}$ to a branching random walk, we find that there exist
explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t - g(t)| \leq
1$ and $|M_t - h(t)| \leq 1$ for all sufficiently large $t$. Furthermore, for
each $n$, we study the final time at which fragments of size $k^{-n}$ exist. In
particular, by relating our branching random walk to a certain point process,
we show that, after suitable rescaling, the laws of these times converge to a
Gumbel distribution as $n \to \infty$.
