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

DOI:
https://doi.org/10.1090/S0002-9939-07-09292-1

Published electronically:
September 25, 2007

MathSciNet review:
2350383

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a quadratic form such that the associated algebraic curve contains a rational point. Here we show that there exists a domain such that for almost all , there exists an infinite sequence of nonzero integer triples satisfying the following two properties: (*i*) For each , is an excellent rational approximation to , in the sense that

*ii*) is a rational point on the curve . In addition, we give explicit values of for which both (

*i*) and (

*ii*) hold, and produce a similar result for a certain class of cubic curves.

**1.**Claude Brezinski,*History of continued fractions and Padé approximants*, Springer Series in Computational Mathematics, vol. 12, Springer-Verlag, Berlin, 1991. MR**1083352****2.**Edward B. Burger,*Exploring the number jungle: a journey into Diophantine analysis*, Student Mathematical Library, vol. 8, American Mathematical Society, Providence, RI, 2000. MR**1774066****3.**Edward B. Burger and Robert Tubbs,*Making transcendence transparent*, Springer-Verlag, New York, 2004. An intuitive approach to classical transcendental number theory. MR**2077395****4.**Carsten Elsner,*On rational approximations by Pythagorean numbers*, Fibonacci Quart.**41**(2003), no. 2, 98–104. MR**1990517****5.**A. Ya. Khintchine,*Continued fractions*, Translated by Peter Wynn, P. Noordhoff, Ltd., Groningen, 1963. MR**0161834**

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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:
https://doi.org/10.1090/S0002-9939-07-09292-1

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