  This paper seeks to test if the large-scale galaxy distribution can be
characterized as a fractal system. Tools appropriate for describing galaxy
fractal structures with a single fractal dimension $D$ in relativistic settings
are developed and applied to the UltraVISTA galaxy survey. A graph of
volume-limited samples corresponding to the redshift limits in each redshift
bins for absolute magnitude is presented. Fractal analysis using the standard
$\Lambda$CDM cosmological model is applied to a reduced subsample in the range
$0.1\le z \le 4$, and the entire sample within $0.1\le z\le 6$. Three
relativistic distances are used, the luminosity distance $d_L$, redshift
distance $d_z$ and galaxy area distance $d_G$, because for data at $z\gtrsim
0.3$ relativistic effects are such that for the same $z$ these distance
definitions yield different values. The results show two consecutive and
distinct redshift ranges in both the reduced and complete samples where the
data behave as a single fractal galaxy structure. For the reduced subsample we
found that the fractal dimension is $D=\left(1.58\pm0.20\right)$ for $z<1$, and
$D=\left(0.59\pm0.28\right)$ for $1\le z\le 4$. The complete sample yielded
$D=\left(1.63\pm0.20\right)$ for $z<1$ and $D=\left(0.52\pm0.29\right)$ for
$1\le z\le6$. These results are consistent with those found by Conde-Saavedra
et al. (2015; arXiv:1409.5409v1), where a similar analysis was applied to a
much more limited survey at equivalent redshift depths, and suggest that either
there are yet unclear observational biases causing such decrease in the fractal
dimension, or the galaxy clustering was possibly more sparse and the universe
void dominated in a not too distant past.
