Fractal and Multi-Fractal Analysis for A Family of Subset Sum Functions: Combinatorial Structures of Embedding Dimension $1$

Abstract

We introduce two frameworks in order to deal with fractal and multi-fractal analysis for subset sum problems where some embedding into the $1$-dimensional Euclidean space plays an important role. As one of these frameworks, the notion of the combinatorial $q$-fractal dimension for a subset sum function is introduced. Thereby, ``non-classical'' generalized dimensions for a family of subset~sum functions can be defined. These generalized dimensions include the box-counting dimension, the information dimension and the correlation dimension as well as the classical case. The combinatorial $q$-fractal dimension includes the density of the subset sum problem. As the other framework, we construct a self-similar set for a particular subset sum function in a family of subset sum functions by using a graph theoretical technique. In this paper, we give a lower bound for a combinatorial $q$-fractal dimension and we show the relations between the three parameters: the number of connected components in a graph, the Hausdorff dimension and a combinatorial $q$-fractal dimension.