Zeta-Dimension

TL;DR

This paper explores the concept of zeta-dimension, a fractal dimension for sets of positive integers, extending its theory and connecting it with classical fractal dimensions and algorithmic information theory.

Contribution

It develops the basic theory of zeta-dimension, introduces new connections with classical fractal dimensions, and presents novel results on its properties and characterizations.

Findings

Extended connections between zeta-dimension and classical fractal dimensions

A gale characterization of zeta-dimension

A theorem on zeta-dimensions of sums and products of sets

Abstract

The zeta-dimension of a set A of positive integers is the infimum s such that the sum of the reciprocals of the s-th powers of the elements of A is finite. Zeta-dimension serves as a fractal dimension on the positive integers that extends naturally usefully to discrete lattices such as the set of all integer lattice points in d-dimensional space. This paper reviews the origins of zeta-dimension (which date to the eighteenth and nineteenth centuries) and develops its basic theory, with particular attention to its relationship with algorithmic information theory. New results presented include extended connections between zeta-dimension and classical fractal dimensions, a gale characterization of zeta-dimension, and a theorem on the zeta-dimensions of pointwise sums and products of sets of positive integers.