From: dmoews@xraysgi.ims.uconn.edu (David Moews)
Newsgroups: sci.math
Subject: Re: Questions about R^2
Date: 10 Sep 1998 20:20:09 -0700

In article <6t9q30$ngq@sjx-ixn5.ix.netcom.com>,
Daniel Giaimo <rgiaimo@nospam.ix.netcom.com> wrote:
|    Is there an infinite non-colinear subset A of R^2 such that if a,b are in 
|A then dist(a,b) is in Q?  Is there such a subset A that is dense?  What about
|the general question for R^n?

If we identify R^2 with the complex plane, we can take A to be 
{(a+ib)^2|a,b rational, a^2+b^2=1}.  This A is a subset of the unit circle.
Whether a set with all distances rational can be dense in the plane is given 
as an open problem in Guy, UNSOLVED PROBLEMS IN NUMBER THEORY; see entry D20.
According to the entry, it is also open whether there exist 7 points in the 
plane with rational distances, no 3 points collinear, and no 4 concyclic.
-- 
David Moews                                   dmoews@xraysgi.ims.uconn.edu