Abstract

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Spectral Geometry of Fractal Information: Complex Dimensions, the Riemann Zeta Distribution, and Algorithmic Oscillations in Sorting and 2D Packing

This chapter presents a comprehensive analysis of the relationship between fractal self-affinity and the Riemann zeta distribution within the context of sorting algorithms and two-dimensional packing problems. We demonstrate that real-world data---ranging from word frequencies in natural language corpora to spatial distributions of geographic features---rarely conforms to classical uniformity assumptions. Instead, such data follows Zipfian (zeta) distributions and exhibits fractal self-affinity, properties that fundamentally alter algorithmic behavior.

Drawing on the theory of complex dimensions developed by Lapidus and van Frankenhuijsen, we establish that the geometric zeta function of a fractal string admits meromorphic continuation, yielding poles that encode not merely a single fractal dimension but an entire spectrum of complex dimensions \( s = \sigma + it \). These complex dimensions manifest as periodic oscillations in geometric quantities such as the tubular neighborhood volume and, crucially, in algorithmic performance metrics including sorting runtime and spatial query costs.

We apply Mellin transform techniques, following Flajolet's methodology, to analyze divide-and-conquer recurrences and demonstrate that the "wobble" in average-case complexity arises from the same spectral structure governing fractal geometry. For two-dimensional packing, we examine the Apollonian gasket as an archetypal fractal arrangement, showing that its Hausdorff dimension \( \delta \approx 1.30568 \) determines the power-law distribution of circle radii. Zagier's theorem establishes a direct connection between the convergence rate of local packing density and the zeros of the Riemann zeta function, implying that the Riemann Hypothesis constrains error bounds in spatial algorithms.

We extend these results to R-tree performance on fractal datasets, incorporating the Faloutsos-Kamel cost model based on correlation dimension. The synthesis reveals a unified spectral framework: the Riemann zeta zeros govern prime distribution error terms; complex dimensions govern fractal string oscillations; and recurrence roots govern algorithmic performance fluctuations. This equivalence suggests a "Fractal Riemann Hypothesis" wherein well-behaved self-similar structures exhibit complex dimensions aligned on a critical line, yielding bounded, regular oscillations and optimal algorithmic stability.

Keywords: complex dimensions, fractal strings, geometric zeta function, Mellin transform, Apollonian gasket, R-tree, Zipf's law, Riemann Hypothesis, spectral geometry