Exploring cosmic homogeneity with the BOSS DR12 galaxy sample
  In this study, we probe the transition to cosmic homogeneity in the Large
Scale Structure (LSS) of the Universe using the CMASS galaxy sample of BOSS
spectroscopic survey which covers the largest effective volume to date, $3\
h^{-3}\ \mathrm{Gpc}^3$ at $0.43 \leq z \leq 0.7$. We study the scaled
counts-in-spheres, $\mathcal{N}(<r)$, and the fractal correlation dimension,
$\mathcal{D}_2(r)$, to assess the homogeneity scale of the universe using a
$Landy\ \&\ Szalay$ inspired estimator.
  Defining the scale of transition to homogeneity as the scale at which
$\mathcal{D}_2(r)$ reaches 3 within $1\%$, i.e. $\mathcal{D}_2(r)>2.97$ for
$r>\mathcal{R}_H$, we find $\mathcal{R}_H = (63.3\pm0.7) \ h^{-1}\
\mathrm{Mpc}$, in agreement at the percentage level with the predictions of the
$\Lambda$CDM model $\mathcal{R}_H=62.0\ h^{-1}\ \mathrm{Mpc}$. Thanks to the
large cosmic depth of the survey, we investigate the redshift evolution of the
transition to homogeneity scale and find agreement with the $\Lambda$CDM
prediction. Finally, we find that $\mathcal{D}_2$ is compatible with $3$ at
scales larger than $300\ h^{-1}\ $Mpc in all redshift bins.
  These results consolidate the Cosmological Principle and represent a precise
consistency test of the $\Lambda CDM$ model.
