Diophantine Approximation Along Algebraic Curves

Edward Burger, Williams College
Let F(x,y) in Z[x,y] be a quadratic form such that the associated curve C: F(x,y)=1 contains a rational point. Here we show that there are many real numbers $\xi$ such that for each one we can find 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 \lim_{n\rightarrow\infty}|\xi y_n-x_n|=0\ ; and (ii) (x_n/z_n,y_n/z_n) is a rational point on the curve C. We will offer some explicit examples and also mention the corresponding issue with cubic curves. This work is joint with Ashok Pillai, an undergraduate from Williams College. Thus any interested undergraduates and graduate students are invited to attend.