Multifractal analysis and lacunarity spectrum of the galaxies of the
  ninth Sloan Digital Sky Survey (SDSS) data release
  In this work, we develop a statistical analysis of the large-scale clustering
of matter in the Universe from the fractal point of view using galaxies from
the Ninth Sloan Digital Sky Survey (SDSS) Data Release (DR9). From the total
set of galaxies, a magnitude-limited sample of galaxies with redshifts in the
range 0 < z < 0.15 was created. The sample covers the largest completely
connected area of the celestial sphere within the catalogue, with limits in
right ascension from 120 to 240 degrees and declination from 0 to 60 degrees,
which is a region that includes the largest galactic samples that have been
studied from the fractal viewpoint to date. The sample contains 164,168
galaxies.
  Using the sliding-window technique, the multifractal dimension spectrum and
its dependence on radial distance are determined. This generalisation of the
concept of fractal dimension is used to analyse large-scale clustering of
matter in complex systems. Likewise, the lacunarity spectrum, which is a
quantity that complements the characterisation of a fractal set by quantifying
how the set fills the space in which it is embedded, is determined.
  Using these statistical tools, we find that the clustering of galaxies
exhibits fractal behaviour that depends on the radial distance for all
calculated quantities. A transition to homogeneity is not observed in the
calculation of the fractal dimension of galaxies; instead, the galaxies exhibit
a multifractal behaviour whose dimensional spectrum does not exceed the
physical spatial dimension for radial distances up to 180 Mpc/h from each
centre within the sample. Our results and their implications are discussed in
the context of the formation of large-scale structures in the Universe.
