Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Choi, Seokhyun
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4414970073
Cites:
Cited by:
Related to: