Dear Dr. Mittleman,
Thanks for your excellent question. Also I hope that the proposal in our essay does not contradict Piero's (i.e. Dr. Nicolini's) and his co-worker's paper since it was Piero who got me thinking about self-similarity and who explained to me some issues with self-similarity and QM/QFT when I visited him last fall. We should have cited the paper you mention and let me recommend this as an interesting and important paper.
That being said I think there is no conflict between our proposal and Piero's paper. In this paper Piero is interested in the self-similarity of a fractal space-time path. The self-similarity we propose is more abstract in that it is a similarity between the forms of the higher order corrections to the action (i.e. I_i ~ I_0 or the ith correction to the action has the same form as the 0th action and alpha_i ~ alpha_1 or the ith coefficients are similar to the 1st coefficient).
However in the essay we do make some connection between this abstract self-similarity and the more usual self-similarity of a path -- as one goes to higher order corrections I_i and alpha_i one is going higher energies and *usually* this means shorter distances. But in QG this is not necessarily the case due to the "UV/IR connection" which is discussed very nicely by Susskind and Lindesay in reference [1] of our essay. Basically the UV/IR connection means that the standard idea "higher energy --> shorter distances" breaks down at some point when gravity enters the picture. The argument very briefly is that at first as one goes to higher energy scales one does probe shorter distances as standard QM implies. However at some very large energy the particles one collides are within each others event horizon and the result of the collision is a BH with some horizon radius. The horizon radius now sets the length scale one can probe. As one increases the energy the horizon grows and thus the length scale one can probe *increases*. Thus our self-similarity is to be taken in the sense of each higher correction probing some higher energy scale. Conventionally this means shorter distance scale in QG this is not necessarily the case due to the UV/IR connection of QG.
Sorry for the long reply and feel free to ask follow up questions/comments. And also thanks for reminding of of Piero's excellent paper which we should have cited.
Best, Doug