Error bounds on the estimation of fractal dimension

The use of fractal dimension as an index of complexity is widely used in natural science. Studies related to the significance of the estimates obtained have rarely been presented in the past. This paper is intended to present an error analysis in the estimation of the fractal dimension of a particular class of objects, namely, graphs of Knopp functions. The algorithm used to estimate the fractal dimension is the variation method. Various numerical approximations of the variation will be presented. Using functional properties of the chosen parametric family of sets, we derive error bounds useful in the assessment of the accuracy that can be achieved in the numerical approximation of the dimension at a given scale and resolution.