  We present a scaling theory of the many-body localisation transition in terms
of emergent, characteristic energyscales. The analysis is based on the
decomposition of the eigenstates in the basis of trivially localised states,
resolved in the energies of the latter, which we refer to as the spectral
decomposition of the eigenstates. The characteristic energyscales emerge when
the multifractal properties, or lack thereof, of the spectral decomposition are
studied at different scales. These characteristic scales correspond to the
ones, above which the spectral decompositions exhibit their global behaviour,
namely full ergodicity in the ergodic phase and multifractality in the
many-body localised phase. On the other hand, at scales below the
characteristic ones, the decomposition in the ergodic phase shows finer
(multi)fractal structures whereas in the localised phase, the decomposition
picks out well-separated, localised resonant peaks. The scaling of these
characteristic energyscales across the many-body localisation transition admits
a scaling theory consistent with a Kosterlitz-Thouless type scenario and bears
striking resemblances to that of inverse participation ratios of eigenstates.
