A Hausdorff dimension for finite sets

TL;DR

This paper introduces a finite Hausdorff dimension for finite sets, which is non-trivial and relates to neighbor proximity, with a convergence theorem linking it to classical Hausdorff dimension.

Contribution

It defines a new finite Hausdorff dimension for finite sets and proves a convergence theorem connecting it to classical Hausdorff dimension.

Findings

Finite Hausdorff dimension is non-trivial for finite sets.

Bound on finite Hausdorff dimension implies local neighbor proximity.

Finite Hausdorff dimension converges to classical Hausdorff dimension under set convergence.

Abstract

The classical Hausdorff dimension of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension which is non-trivial. It turns out that a finite bound for the finite Hausdorff dimension guarantees that every point of the set has "nearby" neighbors. This property is important for many computer algorithms of great practical value, that obtain solutions by finding nearest neighbors. We also define an analog for finite sets of the classical box-counting dimension, and compute examples. The main result of the paper is a Convergence Theorem. It gives conditions under which, if a sequence of finite sets converges to a compact set (convergence of compact subsets of Euclidean space under the Hausdorff metric), then the finite Hausdorff dimension of the finite sets will converge to the classical Hausdorff dimension of the compact set.