Dense set in plane. Michael Spivak problem

Submitted by Structure on Sun, 12/16/2007 - 17:49.

Analysis questions

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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Dense set in plane. Michael Spivak problem