Asymptotically sharp dimension estimates for $k$-porous sets

TL;DR

This paper derives an asymptotically precise upper bound on the Minkowski dimension of k-porous sets in Euclidean space, showing it approaches n-k as porosity increases, thus advancing understanding of fractal geometry with holes.

Contribution

It provides the first asymptotically sharp upper bound for the Minkowski dimension of k-porous sets with holes in multiple directions, refining previous estimates.

Findings

Upper bound tends to n-k as porosity approaches maximum

Establishes sharpness of the bound asymptotically

Connects porosity levels with fractal dimension estimates

Abstract

In $\mathbb{R}^n$, we establish an asymptotically sharp upper bound for the upper Minkowski dimension of $k$-porous sets having holes of certain size near every point in $k$ orthogonal directions at all small scales. This bound tends to $n-k$ as $k$-porosity tends to its maximum value.