Fractal Zeta Functions and Fractal Drums

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This monograph presents a comprehensive and accessible exploration of a new higher-dimensional theory of complex dimensions, applicable to bounded subsets of Euclidean spaces and their generalization, relative fractal drums. It extends the existing theory of zeta functions for fractal strings to fractal and arbitrary bounded sets across any dimension. The introduction of two new classes of fractal zeta functions—distance and tube zeta functions—along with their properties, is a key feature. The theory is developed methodically, with ample examples and historical context to relate the concepts. A significant focus is on the complex dimensions of bounded sets, linking them to Minkowski content and measurability, as well as fractal tube formulas. Notably, it reveals that essential singularities of fractal zeta functions can arise in various fractal sets, influencing their geometry. This work proposes a new definition of fractality based on geometric oscillations and nonreal complex dimensions. The connections to previous research by the first author and collaborators on geometric zeta functions are clearly articulated. The book introduces many novel concepts, presenting numerous open problems and research directions, making it a valuable resource for graduate students and researchers in Fractal Geometry and related fields, while remaining accessible to newcomers.