Spectral Geometry of Fourier Curves with Prime Frequencies: A Comparative Experimental Study

1. Problem Statement

The paper addresses a central question in experimental mathematics: Does the complex, multiscale geometric structure observed in planar curves generated by Fourier series with prime frequencies arise from the specific arithmetic properties of primes, or is it a generic consequence of sparsity and density?

While sparse trigonometric series with arithmetic frequency sets have been studied in analytic number theory, the specific geometric manifestation of the series defined by prime frequencies and factorial valuations had not been rigorously tested against randomized controls. The authors aim to distinguish between "arithmetic structure" and "generic statistical properties" by comparing the prime frequency model against carefully designed randomized alternatives.

2. Methodology

A. The Prime Frequency Model

The core object of study is a finite Fourier series $F_n(t)$ defined for $n \in \mathbb{N}$:
$$F_n(t) = \sum_{p \le n} v_p(n!) e^{ipt}, \quad t \in [-\pi, \pi]$$

• Frequencies: The sum runs over all prime numbers $p \le n$.
• Coefficients: The coefficients are $v_p(n!)$, the $p$-adic valuation of $n!$, calculated via Legendre's formula: $v_p(n!) = \sum_{k \ge 1} \lfloor \frac{n}{p^k} \rfloor$.
• Sampling: The series is sampled at equidistant points $t_j$ in the complex plane to generate a planar point set $\Gamma_n$, interpreted as a polygonal curve.
• Normalization: Curves are centered at their centroid and scaled to a unit diameter to ensure comparability across different $n$.

B. Randomized Control Models

To isolate the role of arithmetic structure, three control models were constructed. All models preserve the multiset of coefficients $W_n = \{v_p(n!) : p \le n\}$ but alter the frequency set or the coupling between frequencies and coefficients:

1. Random Frequency Model: Replaces prime frequencies with a uniformly chosen subset of integers $\{1, \dots, n\}$ of the same cardinality. This preserves sparsity and coefficient magnitudes but destroys the arithmetic structure of the frequency set.
2. Cramér Type Random Model: Generates a frequency set based on independent Bernoulli trials with success probability $1/\log k$ (mimicking the Prime Number Theorem density). This preserves the average density of primes but removes higher-order correlations.
3. Shuffled Prime Model: Retains the prime frequencies but randomly permutes the coefficients $v_p(n!)$ among them. This destroys the structured coupling between specific primes and their factorial valuations while preserving the distribution of both.

C. Geometric Analysis (Box Counting)

The geometric complexity is quantified using box counting:

• The plane is covered by squares of side length $\epsilon$.
• $N(\epsilon)$ is the number of boxes intersecting the curve.
• An effective scaling exponent is defined as $b_d(\epsilon) = \frac{\log N(\epsilon)}{\log(1/\epsilon)}$.
• The study does not claim convergence to a strict Hausdorff dimension but uses $b_d(\epsilon)$ as a descriptive statistic for scale-dependent complexity.
• Ensemble: Results are based on Monte Carlo ensembles (typically $R=200$ realizations) with high sampling density ($N \ge 2^{13}$) to ensure numerical stability.

3. Key Contributions

• Formulation of a Comparative Framework: The paper establishes a rigorous experimental framework to test whether geometric features of arithmetic Fourier series are intrinsic to the number theory or generic to sparse series.
• Definition of Control Models: It introduces three distinct randomized models that systematically decouple frequency density, coefficient distribution, and arithmetic correlation.
• Empirical Evidence of Arithmetic Specificity: It provides the first experimental evidence that the multiscale behavior of these specific Fourier curves is unique to the prime-frequency/factorial-valuation coupling.

4. Experimental Results

The study compared the scaling behavior of the prime model against the control models:

• Prime Frequency Model:

  • Exhibits extended intermediate scaling ranges.
  • Effective exponents $b_d(\epsilon)$ are consistently bounded away from 1 (smooth line) and 2 (space-filling curve).
  • Shows moderate, stable oscillatory variation as $n$ increases, persisting across different sampling densities.

• Random Frequency Model:

  • Exhibits rapid saturation of box counts at small scales.
  • Effective exponents approach 2, indicating a lack of stable intermediate scaling regimes.

• Cramér Type Model:

  • Produces partial multiscale structure.
  • However, it shows reduced variability in $b_d(\epsilon)$ and weaker oscillations compared to the prime model, suggesting that density alone is insufficient to replicate the complexity.

• Shuffled Prime Model:

  • Displays unstable scaling behavior with strong dependence on individual realizations.
  • Loses persistent effective exponents across scales, indicating that the specific pairing of primes and valuations is critical.

5. Significance and Conclusion

The paper concludes that the observed geometric complexity cannot be attributed solely to sparsity or frequency density. Instead, the multiscale behavior arises from the structured coupling between the set of prime frequencies and the factorial valuation weights.

• Implication: The geometric features are a property of the specific Fourier representation built from these arithmetic data, rather than a generic feature of sparse trigonometric series.
• Future Work: The authors motivate further theoretical investigation into:
  • Rigorous bounds on effective scaling exponents.
  • Identifying which specific arithmetic correlations drive the geometric complexity.
  • Exploring whether other additive arithmetic functions yield similar multiscale behaviors.

In summary, this work bridges analytic number theory and spectral geometry, demonstrating that the "shape" of prime numbers, when encoded via Fourier series with factorial weights, possesses a unique geometric signature distinct from random or semi-random approximations.