An overview of complex fractal dimensions: from fractal strings to fractal drums, and back

Abstract

Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with fractal boundary), in \S\ref{Sec:2}, and then in the higher-dimensional case of relative fractal drums and, in particular, of arbitrary bounded subsets of Euclidean space of $\mathbb{R}^N$, for any integer $N \geq 1$, in \S\ref{Sec:3}. Special attention is paid to discussing a variety of examples illustrating the general theory rather than to providing complete statements of the results and their proofs, for which we refer to the books \cite{Lap-vF4} (2013, joint with M. van Frankenhuijsen) when $N=1$, and \cite{LapRaZu1} (2017, joint with G. Radunovi\' c and D. \v{Z}ubrini\'c) when $N \geq 1$ is arbitrary. Finally, in an epilogue (\S\ref{Sec:4}), entitled "From quantized number theory to fractal cohomology", we briefly survey aspects of related work (motivated in part by the theory of complex fractal dimensions) of the author with H. Herichi (in the real case) \cite{HerLap1}, along with \cite{Lap8}, and with T. Cobler (in the complex case) \cite{CobLap1}, respectively, as well as in the latter part of a book in preparation by the author, \cite{Lap10}.