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) [6]). 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, m 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 − 1 . In particular, these sets are used to demonstrate that the inequalities obtained in Rataj and Winter (in press) [6] regarding the general relation of these two dimensions are best possible.