## Dense set in plane. Michael Spivak problem

Posted December 16th, 2007 by Structure

On Cramster.com someone post a question and nobody answer for 6 months.

Here is the question and the answer(mine)

Construct a set A in [0,1]x[0,1] such that A contains at most one point on each horizontal and each vertical line but A is dense in [0,1]x[0,1].(please explain exactly as far as possible)

Here is a set with the desired property.

One idea is to take cartesian dense product QxQ and rotate it around origin with an angle with non rational tangent.

Let

If two points has one common component, say

then

But if then contradiction square-root-2-irational

That is all.

This is one dense set with the desired property.Of course this set is dense in the hole . It is easy to consider only points in