On diophantine approximation along algebraic curves

Abstract:

Let $F(x,y)\in \mathbb {Z}[x,y]$ be a quadratic form such that the associated algebraic curve $\mathcal {C} : F(x,y)=1$ contains a rational point. Here we show that there exists a domain $\mathcal {D} \subseteq \mathbb {R}$ such that for almost all $\xi \in \mathcal {D}$, there exists an infinite sequence of nonzero integer triples $(x_{n},y_{n},z_{n})$ satisfying the following two properties: (i ) For each $n$, $x_{n}/y_{n}$ is an excellent rational approximation to $\xi$, in the sense that \begin{equation*} \lim _{n\rightarrow \infty }| \xi y_{n}-x_{n}|=0\ ; \end{equation*} and (ii ) $(x_{n}/z_{n},y_{n}/z_{n})$ is a rational point on the curve $\mathcal {C}$. In addition, we give explicit values of $\xi$ for which both (i ) and (ii ) hold, and produce a similar result for a certain class of cubic curves.