On diophantine approximation along algebraic curves
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- by Edward B. Burger and Ashok M. Pillai PDF
- Proc. Amer. Math. Soc. 136 (2008), 11-19 Request permission
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.References
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Additional Information
- Edward B. Burger
- Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
- Email: eburger@williams.edu
- Ashok M. Pillai
- Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
- Received by editor(s): August 1, 2006
- Published electronically: September 25, 2007
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 11-19
- MSC (2000): Primary 11J04, 11J70
- DOI: https://doi.org/10.1090/S0002-9939-07-09292-1
- MathSciNet review: 2350383