  Homogeneous mass fragmentation processes describe the evolution of a unit
mass that breaks down randomly into pieces as time. Mathematically speaking,
they can be thought of as continuous-time analogues of branching random walks
with non-negative displacements. Following recent developments in the theory of
branching random walks, in particular the work of \cite{AShi10}, we consider
the problem of the Seneta-Heyde norming of the so-called additive martingale at
criticality. Aside from replicating results for branching random walks in the
new setting of fragmentation processes, our main goal is to present a style of
reasoning, based on $L^p$ estimates, which works for a whole host of different
branching-type processes. We show that our methods apply equally to the setting
of branching random walks, branching Brownian motion as well as Gaussian
multiplicative chaos.
