Lower S-dimension of fractal sets

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. Rataj and Winter (in press) 161). While the upper dimensions always coincide and the upper contents are bounded by each other, the bounds obtained in Rataj and Winter (in press) [6] 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, in with 0 < s < m < 1, we construct sets F in R-d with lower S-dimension s + d - 1 and lower Minkowski dimension m + d I. In particular, these sets are used to demonstrate that the inequalities obtained in Rataj and Winter (in press) 161 regarding the general relation of these two dimensions are best possible. (C) 2010 Elsevier Inc. All rights reserved.