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

TL;DR

This study compares the geometric complexity of Fourier curves with prime frequencies to randomized models, revealing unique multiscale behaviors that suggest an underlying arithmetic structure.

Contribution

It introduces a novel experimental framework for analyzing prime frequency Fourier curves and demonstrates their distinct multiscale geometric properties compared to randomized controls.

Findings

Prime frequency curves exhibit stable, scale-dependent geometric behavior.

Randomized models do not replicate the multiscale features of prime frequency curves.

Results motivate further theoretical exploration of arithmetic influences on curve geometry.

Abstract

We present a comparative experimental study of planar curves arising from a Fourier series whose frequencies are the prime numbers, together with several randomized control models. Starting from the series $F_n(t)=\sum_{p\le n} v_p(n!)\, e^{i p t},~t\in[-\pi,\pi]$, introduced and motivated in a companion work, we investigate the geometric complexity of the associated planar curves obtained by sampling in the complex plane. To test whether the observed multiscale behavior reflects arithmetic structure or can be reproduced as a generic consequence of sparsity or density, we compare the prime frequency model with randomized alternatives, including random frequency sets, a Cram\'er type random model, and a shuffled coefficient model. Using consistent box counting protocols and Monte Carlo ensembles, we observe stable scale dependent behavior for the prime frequency curves that is not reproduced by the randomized models. All results are experimental and are presented as evidence motivating further theoretical investigation.