Zeta-Dimension

The zeta-dimension of a set A of positive integers is Dimζ(A) = inf{s | ζA(s) < ∞}, where ζA(s) = ∑ n∈A n. Zeta-dimension serves as a fractal dimension on Z that extends naturally and usefully to discrete lattices such as Z, where d is a positive integer. 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.