There is the theory of elliptic curves over Q. Such have to have at least one rational point, sometimes have finitely many, sometimes have infinitely many. When there are infinitely many the component containing the identity is dense with rationals, but the other component (if any) may have no rationals. Not your question, but closely related. On Mon, Sep 16, 2013 at 10:39 AM, Cordwell, William R <wrcordw@sandia.gov>wrote:
Hello,
There was some interesting discussion about rational points being dense in the unit circle (corresponding to all the Pythagorean triples). It is straightforward to show that there are many circles that contain no rational points.
So, is there any theory about (smooth, closed, whatever modifier you like) curves in the plane that completely miss a dense subset?
Thanks, Bill
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