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I was wondering if the notion of a quantity having complex dimension makes sense mathematically? It's maybe just a bad question; I can't attach any physical meaning or intuition to, for example, a vector space of dimension 2 + i, or a quantity with length-dimension -i, but I was wondering if this concept has any mathematical meaning or use.

I did some Google searching, but all I could see were links about complex vector spaces and the like, and I'm not even sure where I should begin looking to find something like this in a textbook or paper... I did come across one post on this site

Is there a complex analogue of the covering dimension?

which mentions complex dimensions in passing, but it has no answer and I hardly understand the question. I will continue digging into the tags once I finish posting.

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Slogan: the "dimensions" of a geometric object $X$ are the poles of its zeta function $\zeta_X(s)$ (whatever that is). This idea works pretty well for algebraic curves and surfaces.

In the past couple of decades, it's been shown to work for bounded sets in $\mathbb{R}^N$, including fractals. In this setting, having a "complex dimension" means that, when the radius of the balls covering the set scale by $\lambda$ , the volume of the balls scales by about $\lambda^af(\ln \lambda)$, where $a$ is related to the real part of the dimension and $f$ is periodic with a period related to the imagniary part of the dimension.

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