Scaling properties of discrete fractals

TL;DR

This paper investigates how finite size and discreteness affect the self-similar properties of discrete fractal samples, providing estimates for reliable measurement scales, with implications for analyzing galaxy distributions.

Contribution

It introduces a method to estimate the scale above which statistical properties of discrete fractals are reliable, accounting for finite size effects.

Findings

Finite size effects influence fractal self-similarity

An explicit estimate of reliable measurement scale is provided

Results are relevant for galaxy distribution analysis

Abstract

An important problem in the analysis of experimental data showing fractal properties, is that such samples are composed by a set of points limited by an upper and a lower cut off. We study how finite size effect due to the discreteness of the sets may influence self similar properties even far from these cut-offs. Estimations of these effects are provided on the basis of the characteristics of the samples. In particular we present an estimate of the length scale above which information about average quantities is reliable, by explicitly computing discreteness effects in number counting . The results have particular importance in the statistical analysis of the distribution of galaxies.