Comment on the paper ``The ESO Slice Project galaxy redshift survey: V. Evidence for a D=3 sample dimensionality''

In a recent analysis of number counts in the ESP survey Scaramella et al. (1998) claim to find evidence for a cross-over to homogeneity at large scales, and against a fractal behaviour with dimension $D \approx 2$. In this comment we note firstly that, if such a cross-over exists as described by the authors, the scale characterizing it is ~ 100 - 300 Mpc/h. This invalidates the ``standard'' analysis of the same catalogue given elsewhere by the authors which results in a ``correlation length'' of only r_0 = 4 Mpc/h. Furthermore we show that the evidences for a cross-over to homogeneity rely on the choice of cosmological model, and most crucially on the so called K corrections. We show that the D ~ 3 behaviour seen in the K-corrected data of Scaramella et al. is in fact unstable, increasing systematically towards D=4 as a function of the absolute magnitude limit. This behaviour can be quantitatively explained as the effect of an unphysical K-correction in the relevant range of red-shift (z ~ 0.1- 0.3). A more consistent interpretation of the number counts is that D is in the range 2 - 2.5, depending on the cosmological model, consistent with the continuation of the fractal D ~ 2 behaviour observed at scales up to ~100 Mpc/h. This implies a smaller K-correction. Given, however, the uncertainty in the effect of intrinsic fluctuations on the number counts statistic, and its sensitivity on these large scales to the uncertain K corrections, we conclude that it is premature to put a definitive constraint on the galaxy distribution using the ESP data alone.