2 Yadav, Bagla and Khandai 2 FRACTALS AND FRACTAL DIMENSIONS to be expected as we see only brighter galaxies at larger distances in a flux or magnitude limited sample. For a given survey one computes the fractal dimension of the point set under consideration and the scale beyond which the fractal dimension is equal to the ambient dimension of the space in which the particles are distributed is referred to as the scale of homo- geneity of that distribution. Mathematically the fractal dimension is defined for an infinite set of points. Given that the observa tional samples are finite there is a need to understand the deviation s in the fractal dimensions arising due to the finite number of poi nts. In practice we should identify a scale to be the scale of homogeneity where the rms error in the fractal dimension is comparable to or greater than the deviation of the fractal dimension from the ambi- ent dimension (Bagla et al. 2008). It is not possible to distinguish between a given point set and a homogeneous point set with the same number of points in the same volume beyond the scale of homogeneity . In all practical cases of interest the fractal dimension is never equal to the ambient integer dimension of the space. Bagla et al. (2008) have shown that these deviations in fractal dimension oc- cur mainly due to weak clustering present in the galaxy distribu- tion, with a smaller contribution arising due to the finite nu mber of galaxies in the distribution. This work assumed the standard cos- mological model in order to derive a relation between the fractal dimension on one hand, and, a combination of number density of tracers of the density field and clustering on the other. It wa s as- sumed that the clustering is weak at scales of interest. In this paper, we revisit the expected deviation of fractal dimension for a finite distribution of weakly clustered points and verify the relations de- rived in Bagla et al. (2008). For this purpose we have used a parti- cle distribution from a large volume N − Body simulations. Further, we generalise the relation between clustering and fractal dimension to compute the rms error in determination of fractal dimensions and clustering from data. We then discuss the implications of this for the expected scale of homogeneity in the standard cosmologi- cal model. We also comment on some recent determinations of the scale of homogeneity from observations of galaxy distribution in redshift surveys. The plan of the paper is as follows. In section § 2 we present a brief introduction to fractals and fractal dimensions. Subsection § 2.1 describes the fractal dimension for a weakly clustered distri- bution of finite number of points. Section § 3 discusses the calcu- lation of variance in ξ and hence the variance in offset to fractal dimension. We present the results in section § 4, and conclude with a summary of the paper in § 5. The name fractal was introduced by Benoit B. Mandelbrot (Mandelbrot 1982) to characterise geometrical figures whic h may not be smooth or regular. One of the definitions of a fractal isthat it is a shape made of parts similar to the whole in some way. It is use- ful to regard a fractal as a set of points that has properties such as those listed below, rather than to look for a more precise defi nition which will almost certainly exclude some interesting cases. A set F is a fractal if it satisfies most of the following (Falconer 20 03): where Θ( x ) is the Heaviside function. If we consider the distribu- tion function for the number of points inside such spherical cells, we can rewrite C 2 in terms of the probability of finding n particles in a sphere of radius r . Here we assume that we have N points in the distribution. M is the number of centres on which spheres of radius r have been placed with the requirement that the entire sphere lies inside the distribu- tion of points. Spheres are centred on points within the point set and it is clear that M < N . Here n i ( r ) denotes the number of particles within a distance r from i th point: we define D$_{b}$ to be the box counting dimension (Barab´asi & Stanley 1995). In general D$_{b}$ is a function of scale. One of the difficulties with such a definition is that it does not depend on the number of particles inside the boxes and rather depends only on the number of boxes. It provides very limited information about the degree of clumpiness of the distribution and is more of a geometrical measure. To get more detailed information on clustering of the distribution we use the concept of correlation dimension. We have chosen the correlation dimension, among various other definitions (see e.g. Borgani 1995; Mart´ınez & Saar 2002), due to its mathematical simplicity and the ease with which it can be adapted to calculations for a given point set. The formal definition of correlation dimension demands that the numberof particles in the distribution should be infinite (e.g. N → ∞ in equation 2). We use a working definition for a finite point set. Calculation of the correlation dimension requires the introduction of correlation integral given by The simplest definition of the fractal dimension is the so cal led Box counting dimension . Here we place a number of mutually ex- clusive boxes that cover the region in space containing the point set and count the number of boxes that contain some of the points of the fractal. The fraction of non-empty boxes clearly depends on the size of boxes. Box counting dimension of a fractal distribution is defined in terms of non empty boxes N ( r ) of radius r required to cover the distribution. If Fractal dimension provides a description of how much space a point set fills. It is also a measure of the prominence of the ir regu- larities of a point set when viewed at a given scale. We may regard the dimension as an index of complexity. We can, in principle, use the concept of fractal dimensions to characterise any point set. Fractals are characterised using the so called fractal dimensions. These can be defined in different ways, which do not necessari ly coincide with one another. Therefore, an important aspect of study- ing a fractal structure is the choice of a definition for fract al dimen- sion that best applies to the case in study. C 2 ( r ) = 1 N N ∑ n =0 nP ( n ; r,N ) (4) n i ( r ) = N ∑ j =1 Θ( r − | x i − x j | ) (3) C 2 ( r ) = 1 NM M ∑ i =1 n i ( r ) (2) N ( r ) ∝ r D$_{b}$ (1) (ii) F is too irregular to be described in traditional geometrical language, both locally and globally. (i) F has a fine structure, i.e., detail on arbitrarily small scale s. (iii) F has some form of self-similarity, perhaps approximate or statistical. (iv) In most cases of interest F is defined in a very simple way, perhaps recursively.