Multifractal spectra and precise rates of decay in homogeneous fragmentations

We consider a mass-conservative fragmentation of the unit interval. Motivated by a result of Berestycki [J. Berestycki, Multifractal spectra of fragmentation processes, J. Statist. Phys. 113 (3-4) (2003) 411-430], the main purpose of this work is to specify the Hausdorff dimension of the set of locations having exactly an exponential decay. The study relies on an additive martingale which arises naturally in this setting, and a class of Lévy processes constrained to stay in a finite interval.