The claim that the large scale structure of the Universe is heirarchical has a very long history going back at least to Charlier ’ s papers of the early 20th century .
In recent years , the debate has centered largely on the works of Sylos Labini , Joyce , Pietronero and others , who have made the quantative claim that the large scale structure of the Universe is quasi-fractal with fractal dimension D \approx 2 .
There is now a concensus that this is the case on medium scales , with the main debate revolving around what happens on the scales of the largest available modern surveys .

Apart from the ( essentially sociological ) problem that their thesis is in absolute conflict with any concept of a Universe with an age of \approx 14 billion years or , indeed , of any finite age , the major generic difficulty faced by the proponents of the heirarchical hypothesis is that , beyond hypothesizing the case ( eg : Nottale ’ s Scale Gravity ) , there is no obvious mechanism which would lead to large scale structure being non-trivially fractal .
This paper , which is a realization of a worldview that has its origins in the ideas of Aristotle , Leibniz , Berkeley and Mach , provides a surprising resolution to this problem : in its essence , the paper begins with a statement of the primitive self-evident relationship which states that , in the universe of our experience , the amount of material , M , in a sphere of redshift radius R _ { z } is a monotonic increasing function of R _ { z } .
However , because the precise relationship between any Earth-bound calibration of radial distance and R _ { z } is unknowable then fundamental theories can not be constructed in terms of R _ { z } , but only in terms of a radial measure , R say , calibrated against known physics .
The only certainty is that , for any realistic calibration , there will exist a monotonic increasing relationship between R _ { z } and R so that we have M = f ( R ) for a monotonic increasing function f .
But the monotonicity implies R = f ^ { -1 } ( M ) \equiv g ( M ) which , in the absence of any prior calibration of R , can be interpreted as the definition of the radius of an astrophysical sphere in terms of the amount of mass it contains - which is the point of contact with the ideas of Aristotle , Leibniz , Berkeley and Mach .
The development of this idea , and the resulting implications for the geometrical structure of physical space , leads necessarily to the final result , which is that large scale structure in the Universe of our experience is fractal of dimension D = 2.