The scale of homogeneity in the local Universe with the ALFALFA
  catalogue
  We use the scaled counts in spherical caps $\mathcal{N}(<\theta)$ and the
fractal correlation dimension $ \mathcal{D}_{2}(\theta) $ procedures to search
for a transition scale to homogeneity in the local universe as given by the
ALFALFA catalogue (a sample of extragalactic HI line sources, in the redshift
range $0 < z < 0.06$). Our analyses, in the 2-dimensional sky projected data,
show a transition to homogeneity at $\theta_H = 16.49^{\circ} \pm
0.29^{\circ}$, in remarkable accordance with the angular scale expected from
simulations, a result that strengthens the validity of the cosmological
principle in the local universe. We test the robustness of our results by
analysing the data sample using three versions of the $\mathcal{N}(<\theta)$
estimator, which show a well agreement between them. These statistical
estimators were validated using mock realizations generated assuming a fractal
distribution of points, successfully recovering the input information. In
addition, we perform further analyses showing that our approach is also able to
indicate the presence of under- and over-densities in the data. Finally, we
verify the influence of the sample size to the $\theta_H$ estimates by using
segment Cox processes of different projected areas, confirming the suitability
of the ALFALFA surveyed area for the current analyses.
