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Dense set in plane. Michael Spivak problem

Posted December 16th, 2007 by Structure
in

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
$ D=\{(x-y\sqrt 2,x\sqrt 2+y)|(x,y)\in Q\times Q\} $
If two points has one common component, say

$$x_1-y_1\sqrt 2=x_2-y_2\sqrt 2$$

then

$$x_1-x_2=(y_1-y_2)\sqrt 2$$

But if $ y_1-y_2\neq 0 $ then $ \sqrt 2\in Q $ 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 $ R^2 $. It is easy to consider only points in $ [0,1]\times[0,1] $

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