Additive Rigidity for $x$-Coordinates of Rational Points on Elliptic Curves
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910240165855232 |
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| author | Choi, Seokhyun |
| author_facet | Choi, Seokhyun |
| contents | We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$ be an integer, and $0<ρ\leq 1$. We show that if a $d$-dimensional proper generalized arithmetic progression in $\mathbb{Q}$ contains the $x$-coordinates of rational points on $E/\bbq$ with positive proportion $ρ$, then the number of such points is bounded by $A(E,d,ρ)^r$. The proof combines extraction lemmas, gap principles, and the bounds for spherical codes. As an application, we obtain restrictions on sets of rational points whose $x$-coordinates have small sumsets or large additive energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03828 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive Rigidity for $x$-Coordinates of Rational Points on Elliptic Curves Choi, Seokhyun Number Theory 11G05 We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$ be an integer, and $0<ρ\leq 1$. We show that if a $d$-dimensional proper generalized arithmetic progression in $\mathbb{Q}$ contains the $x$-coordinates of rational points on $E/\bbq$ with positive proportion $ρ$, then the number of such points is bounded by $A(E,d,ρ)^r$. The proof combines extraction lemmas, gap principles, and the bounds for spherical codes. As an application, we obtain restrictions on sets of rational points whose $x$-coordinates have small sumsets or large additive energy. |
| title | Additive Rigidity for $x$-Coordinates of Rational Points on Elliptic Curves |
| topic | Number Theory 11G05 |
| url | https://arxiv.org/abs/2510.03828 |