Rigorous Geometric Obstructions for Fourier Curves Generated by Prime Numbers
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917293353598976 |
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| author | Vartziotis, Dimitris |
| author_facet | Vartziotis, Dimitris |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21270 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigorous Geometric Obstructions for Fourier Curves Generated by Prime Numbers Vartziotis, Dimitris General Mathematics 42A16, 11A25, 28A75 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. |
| title | Rigorous Geometric Obstructions for Fourier Curves Generated by Prime Numbers |
| topic | General Mathematics 42A16, 11A25, 28A75 |
| url | https://arxiv.org/abs/2602.21270 |