Systematic error of dimension estimates using fixed mass scaling methods

Methods to estimate the number of degrees of freedom of chaotic dynamical systems suffer from intrinsic errors. The errors are due to the finite extent of phase space and are felt at any finite number of phase space points. We compare the errors of two methods to extract dimensions from scaling properties. One is based on the scaling of the number of points in spheres with varying radius and the other one concerns the scaling of the radius of spheres that contain a varying number of points. We argue that the latter method is preferable and we derive an analytic expression for the error. We compare both this systematic error and the error due to statistical fluctuations in different realizations of random sets to the results of numerical simulations.