I am trying to draw the following set
$$E=\{(x,y)\in\Bbb R^2:x+y,x-y\in F\}$$ where $F$ is the middle third Cantor set:
I think this is what the first iteration looks like:
but I am really not sure (I'm sorry for the pathetic quality by the way, I was forced to use Paint - I will happily create a nicer image if it is the correct one).
Could someone tell me if I am correct and if not, what it should llook like please? Thank you.
The region bounded by the lines $x+y=0$, $x+y=1$, $x-y=0$ and $x-y=1$ is a diamond (rotated square) with opposite corners at $(0,0)$ and $(1,0)$. The zeroth iteration of $E$ is simply that filled-in diamond.
$x+y$ and $x-y$ are perpendicular, so further iterations of $E$ are the intersection of two Cantor sets extended into stripes running across the diamond. Indeed, the first iteration of $E$ should look like this:
$E$ itself is known as Cantor dust, and looks like the below image (rotated of course):
