  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.
