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What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all the euclidean distances from X to points of S are rational?

Can we do better then $3\leq K \leq 2^{\aleph_0}$?

What can be said about K if all the points of S must be at integer distance from eachother, can it be proved to be finite?

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Three points suffice. Take $S′=\{(0,0),(1,0)\}$. Then any point with algebraic distances to $S'$ has algebraic coordinates (it lies on the intersection of circles with algebraic coefficients). Let $S=S'\cup\{(x,y)\}$, where $1,x,y$ are algebraically independent over $\mathbb{Q}$. –  Boris Bukh Mar 7 '12 at 14:07
    
Is there any set of three points that does not have this property? –  Emil Jeřábek Mar 7 '12 at 15:45
    
@Emil Right, for the second question we have $4\leq K$, atleast if they are not colinear. –  Holowitz Mar 7 '12 at 16:01
    
@Emil: Yes. If each of the sides of a triangle has rational length, then the set of points at rational distance from all three of its vertices is dense in the plane. –  Tony Huynh Mar 7 '12 at 16:10
    
@Tony: I can’t say I see this. Is there a simple argument I am missing? –  Emil Jeřábek Mar 7 '12 at 16:37
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1 Answer

up vote 3 down vote accepted

As Boris Bukh points out, three points suffice, but I'd like to point out that your question is related to this MO question.

Here is a summary of the information in the previous question. For the second part of your question, the author (me) conjectures that for any finite set $S$ with all rational distances, no such point $P$ exists. As I noted in the comments, this is true when $|S|=3$, proven by Almering.

It is not known if there is a point with all rational distances to the unit square. However, it is known that there are no points at rational distance from all vertices of a regular $n$-gon, except perhaps when $n=4,6,8,12,24$.

Some more tangential remarks are that it is not known if there is a dense set of points in the plane with all distances rational, although it is conjectured that there is none.

Even more tangential, it is not known if every planar graph can be straight-line embedded in the plane with all edges having rational length, although it is conjectured to be true.

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