Now, I know as much about this as I do brain surgery, but according to that paper the idea is one gets a vacuum "refractive index" due to quantum space-time fluctuations of:

r ~ 1 - (E/M_QGn)^n, where n looks like some quantum number thingy. Why that is 1 minus and not 1 plus I don't know, but that's how they put it, and the M_QGn's are the mass threshold.

The say their results constrain M_QG1 > 0.4 *10^18 GeV, and the
M_QG2 > 0.4 *10^11 GeV. That's still pretty high. I don't understand the 'n' progression, and why it would get lower.

It says string theories and some extra dimensional theories will let the M_QG's be less than the Planck mass.

-Richard