Lower S-Dimension of Fractal Sets

TL;DR

This paper demonstrates that for certain fractal sets, the lower S-dimension can be strictly smaller than the lower Minkowski dimension, showing the bounds are optimal and highlighting the flexibility in fractal dimension relations.

Contribution

It constructs specific sets in R^d to show the lower S-dimension can be strictly less than the lower Minkowski dimension, confirming the bounds are sharp.

Findings

Lower S-dimension can be strictly smaller than lower Minkowski dimension.

Constructed sets demonstrate the optimality of previously established bounds.

The results clarify the relationship between different fractal dimensions.

Abstract

The interrelations between (upper and lower) Minkowski contents and (upper and lower) surface area based contents (S-contents) as well as between their associated dimensions have recently been investigated for general sets in R^d (cf. [3]). While the upper dimensions always coincide and the upper contents are bounded by each other, the bounds obtained in [3] suggest that there is much more flexibility for the lower contents and dimensions. We show that this is indeed the case. There are sets whose lower S-dimension is strictly smaller than their lower Minkowski dimension. More precisely, given two numbers s, m with 0 < s < m < 1, we construct sets in R^d with lower S-dimension s+d-1 and lower Minkowski dimension m+d-1. In particular, these sets are used to demonstrate that the inequalities obtained in [3] regarding the general relation of these two dimensions are best possible.