# The distribution of multiples of real points on an elliptic curve

## TL;DR

This paper studies how multiples of a real point on an elliptic curve are distributed in the plane, providing bounds and density results that connect to classical Diophantine approximation theories.

## Contribution

It introduces bounds on coordinates of multiples and determines their natural density within open subsets, linking elliptic curve point distribution to Diophantine approximation.

## Key findings

- Bounds on x and y coordinates of nP
- Density of nP in open subsets of R^2
- Connection to classical Diophantine approximation

## Abstract

Given an elliptic curve $E$ and a point $P$ in $E(\mathbb{R})$, we investigate the distribution of the points $nP$ as $n$ varies over the integers, giving bounds on the $x$ and $y$ coordinates of $nP$ and determining the natural density of integers $n$ for which $nP$ lies in an arbitrary open subset of $\mathbb{R}^2$. Our proofs rely on a connection to classical topics in the theory of Diophantine approximation.