The complex dimension is defined on 'fractal strings', 'fractal drums/sprays' and 'relative fractal drums'.

- Fractal strings are a set of separated intervals on the number line, e.g. ___ ___ __ _ _ the spacing, location and order of the pieces is unimportant.

The dimension of this as a set is 1, but they define the strings by their boundary (two end points), so all possible fractal strings have a dimension between 0 and 1.

- Fractal drums/sprays seem to be just the same thing but in higher dimensions, e.g. in 2D if could be a set of separate disks or filled squares. Again, they define their dimension just on the boundary, so in 2D I imagine the fractal dimension is between 1 and 2D.

- Relative fractal drums are something like a fractal drum where the boundary (usually a perimeter line in the 2D case) is itself a fractal string.

Anyway, the complex dimension is based on extending the Dirilicht series used to define the string length reduction rate, to the continuous Riemann zeta function. The poles of this function are the dimensions of the object, which can be complex.

It looks like (as in my OP) complex dimensions indicate an oscillation in the growth rate:

In this book, we develop a theory of complex dimensions of fractal strings (i.e., one-dimensional drums with fractal boundary). These complex dimensions are deﬁned as the poles of the corresponding zeta function. They describe the oscillations in the geometry or the frequency spectrum of a fractal string by means of an explicit formula. Such oscillations are not observed in smooth geometries.

What I don't understand is how they apply this 'complex dimension' formula to normal fractal sets. But it appears that they can and have done because the author talks about applying it to basic fractals like the Koch snowflake and Menger carpet.

What surprises me is that complex dimensions implies dimensions can be negative. But they haven't shown any examples of such a shape, and the only mention of supporting negative dimensions applies to the quite odd 'relative fractal drum', in which the negativity represents smoothness in some sense.

Anyway, I'm still struggling with what is motivating these hundreds of pages of work. The author's descriptions are

*crazy*:

the theory of complex dimensions aims at unifying many aspects of fractal and arithmetic geometries. e.g. the notion of fractal membrane (i.e., quantized fractal string), a still conjectural (noncommutative) ﬂow on the moduli space of fractal membranes and correspondingly, a ﬂow of zeta functions (or partition functions) and a ﬂow of zeros are used in an essential manner in order to provide a possible new approach to the Riemann hypothesis. The ﬂow of fractal membranes can be viewed as transforming generalized quasicrystals into (self-dual) pure crystals. Conjecturally, this would also provide a proof of the generalized Riemann hypothesis (GRH), whether it be viewed as a (noncommutative) Ricci-type ﬂow or, physically, as a renormalization ﬂow (not unlike the one which presumably describes the time evolution of our universe).

FYI the Riemann hypothesis is a famous unsolved problem, a Millenium prize problem and one of very few still unsolved problems from Hilbert's list in the 1920s.

This quote might get to his motivation better:

our deﬁnition of fractality cannot just be applied to standard geometric objects but is also applicable, in principle, to ‘virtual’ geometries, spectral geometries, dynamical systems, algebraic and noncommutative geometries as well as arithmetic geometries. In fact, along with complex dimensions, it should be used as a unifying tool between these apparently vastly different domains of mathematics. This long-term goal has been one of the central motivations of the author (and his collaborators)

His definition of 'fractality' is basically that you can call something a fractal if it has at least one non-real complex dimension (with positive real part). He claims it encompasses the set of fractals better than any other definition.

(If this is true, it implies that ALL standard fractals have an imaginary component to their dimension)