Complex Dimensions of Fractal Strings and Oscillatory Phenomena in Geometry andArithmetic

摘要

In this work, the authors develop a theory of 'complex dimensions' of fractalstrings. The authors first establish pointwise and distributional explicit formulas, which should be considered as the basic tools of this theory. Then they apply these explicit formulas to construct the 'spectral operator', that expresses the spectrum in terms of the geometry of a fractal string. The authors also study in detail the complex dimensions of self-similar fractal strings. Further, the authors derive an explicit formula for the volume of the tubular neighborhoods of the boundary of a fractal string. The authors deduce a new criterion for the Minkowski measurability of a fractal string, in terms of its complex dimensions. In the latter part of the geometry and the spectrum of fractal strings. The authors end this paper by proposing as a new definition of fractality the presence of nonreal complex dimensionals.