Lorentz violations in multifractal spacetimes
  Using the recent observation of gravitational waves (GW) produced by a
black-hole merger, we place a lower bound on the energy above which a
multifractal spacetime would display an anomalous geometry and, in particular,
violations of Lorentz invariance. In the so-called multifractional theory with
$q$-derivatives, we show that the deformation of dispersion relations is much
stronger than in generic quantum-gravity approaches (including loop quantum
gravity) and, contrary to the latter, present observations on GWs can place
very strong bounds on the characteristic scales at which spacetime deviates
from standard Minkowski. The energy at which multifractal effects should become
apparent is $E_*>10^{14}\,{\rm GeV}$ (thus improving previous bounds by 12
orders of magnitude) when the exponents in the measure are fixed to their
central value $1/2$. We also estimate, for the first time, the effect of
logarithmic oscillations in the measure (corresponding to a discrete spacetime
structure) and find that they do not change much the bounds obtained in their
absence, unless the amplitude of the oscillations is fine tuned. This feature,
unavailable in known quantum-gravity scenarios, may help the theory to avoid
being ruled out by gamma-ray burst (GRB) observations, for which $E_*>
10^{17}\,{\rm GeV}$ or greater.
