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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On diophantine approximation along algebraic curves


Authors: Edward B. Burger and Ashok M. Pillai
Journal: Proc. Amer. Math. Soc. 136 (2008), 11-19
MSC (2000): Primary 11J04, 11J70
Published electronically: September 25, 2007
MathSciNet review: 2350383
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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

$\displaystyle \lim _{n\rightarrow \infty }\vert \xi y_{n}-x_{n}\vert=0 ; $

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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09292-1
PII: S 0002-9939(07)09292-1
Received by editor(s): August 1, 2006
Published electronically: September 25, 2007
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society