Fractal analysis of the galaxy distribution in the redshift range 0.45 <
  z < 5.0
  Evidence is presented that the galaxy distribution can be described as a
fractal system in the redshift range of the FDF galaxy survey. The fractal
dimension $D$ was derived using the FDF galaxy volume number densities in the
spatially homogeneous standard cosmological model with $\Omega_{m_0}=0.3$,
$\Omega_{\Lambda_0}=0.7$ and $H_0=70 \; \mbox{km} \; {\mbox{s}}^{-1} \;
{\mbox{Mpc}}^{-1}$. The ratio between the differential and integral number
densities $\gamma$ and $\gamma^\ast$ obtained from the red and blue FDF
galaxies provides a direct method to estimate $D$, implying that $\gamma$ and
$\gamma^\ast$ vary as power-laws with the cosmological distances. The
luminosity distance $d_{\scriptscriptstyle L}$, galaxy area distance
$d_{\scriptscriptstyle G}$ and redshift distance $d_z$ were plotted against
their respective number densities to calculate $D$ by linear fitting. It was
found that the FDF galaxy distribution is characterized by two single fractal
dimensions at successive distance ranges. Two straight lines were fitted to the
data, whose slopes change at $z \approx 1.3$ or $z \approx 1.9$ depending on
the chosen cosmological distance. The average fractal dimension calculated
using $\gamma^\ast$ changes from $\langle D \rangle=1.4^{\scriptscriptstyle
+0.7}_{\scriptscriptstyle -0.6}$ to $\langle D \rangle=0.5^{\scriptscriptstyle
+1.2}_{\scriptscriptstyle -0.4}$ for all galaxies, and $D$ decreases as $z$
increases. Small values of $D$ at high $z$ mean that in the past galaxies were
distributed much more sparsely and the large-scale galaxy structure was then
possibly dominated by voids. Results of Iribarrem et al. (2014,
arXiv:1401.6572) indicating similar fractal features with $\langle D \rangle
=0.6 \pm 0.1$ in the far-infrared sources of the Herschel/PACS evolutionary
probe (PEP) at $1.5 \lesssim z \lesssim 3.2$ are also mentioned.
