Part III: Complex Dimensions and Spectral Geometry¶
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Status: chapter draft
Part III is an unreviewed chapter draft. It is a theoretical synthesis that reports no new data; its reference list records web sources accessed in December 2025.
Spectral Geometry of Fractal Information: Complex Dimensions, the Riemann Zeta Distribution, and Algorithmic Oscillations in Sorting and 2D Packing
Keywords: complex dimensions, fractal strings, geometric zeta function, Mellin transform, Apollonian gasket, R-tree, Zipf's law, Riemann Hypothesis, spectral geometry
Real-world inputs to algorithms, from word frequencies to the positions of geographic features, follow Zipfian (Riemann zeta) distributions and are fractally self-affine rather than uniform. This chapter uses the Lapidus and van Frankenhuijsen theory of complex dimensions to show that the geometric zeta function of a fractal string has poles \( s = \sigma + it \) whose imaginary parts appear as log-periodic oscillations in tube volumes, in the Mellin-transform analysis of divide-and-conquer recurrences developed by Flajolet, and in the cost of spatial queries. The Apollonian gasket, with Hausdorff dimension \( \delta \approx 1.30568 \), provides the archetypal two-dimensional packing, and Zagier's theorem ties the convergence of its local packing density to the zeros of the Riemann zeta function. The chapter closes with the Faloutsos-Kamel fractal-dimension cost model for R-trees and a conjectured Fractal Riemann Hypothesis under which the complex dimensions of well-behaved self-similar structures align on a critical line and algorithmic oscillations stay bounded.