Rigorous Geometric Obstructions for Fourier Curves Generated by Prime Numbers

TL;DR

This paper investigates the geometric properties of Fourier curves generated by prime numbers, establishing rigorous obstructions to their regularity and providing insights into their complex behavior.

Contribution

It introduces new mathematical obstructions to the regularity of prime-based Fourier curves and explains their complex geometric behavior with rigorous proofs.

Findings

Curve lengths grow unbounded as n increases

Derivatives of the curves are not uniformly bounded

Diameters grow at least on the order of n log log n

Abstract

We study planar curves defined by finite Fourier series of the form $F_n(t)=\sum_{p\le n} v_p(n!)\, e^{i p t}$, where the frequencies are the prime numbers and $v_p(n!)$ denotes the exponent of the prime $p$ in the factorization of $n!$. We establish several rigorous obstructions to uniform geometric regularity as $n\to\infty$. In particular, we prove that the curve lengths grow without bound, that neither the first nor the second derivatives remain uniformly bounded, and that the diameters grow at least on the order of $n\log\log n$. As a consequence, the covering numbers of the curves satisfy explicit quantitative lower bounds. These results provide a rigorous explanation for the complex geometric behavior observed in numerical investigations of this model.