# Is there a name for subsets $X$ of the domain of a function $f$, especially $f \in SO(n,\Bbb R)$, such that $f(X)$ strictly contains $X$?

I am asking for a friend who is interested in finding subsets $$X$$ of $$\Bbb R^n$$ such that for some rotation $$r$$ one has $$X \subsetneq r(X)$$. Perhaps such things have been looked at and have a name? Any references would be appreciated. Of course, such subsets and rotations exist. But what are they called?

• Just interested in $X$ being a proper subset of $r(X)$. Or $r(X)$ being a proper subset of $X$, if you multiply both sides by $r^{-1}$. Commented Feb 17, 2022 at 18:37
• That is very weird. My mind doesn't even want to imagine sets/maps like this. The image of a set always has lesser or equal cardinality than the set itself. Some tricks like dividing by two the even numbers in order to get the whole set of integers come to mind. But not for simple maps like rotations. They don't have a name that people know of, that's for sure. Commented Feb 17, 2022 at 18:52
• Assuming the Axiom of Choice, the Hausdorff-Banach-Tarski paradox provides lots of such $X, Y$ and $f$. These do not have special names though. Commented Feb 17, 2022 at 19:00
• I realized later that homotethies in vector spaces provide many such examples. Still doubt that rotations have any such set. Commented Feb 18, 2022 at 8:27
• Find a point $x$ and rotation $r_0$ such that the points $r_0^n(x)$, $n \in \Bbb{Z}$, are distinct. Then consider the set $X = \{ r_0^n(x) \vert n = 0, 1, 2,\cdots \}$ . Apply any of the rotations $r=r_0^{-k}$ where $k$ is any positive integer to $X$. Shades of Hilbert's Hotel! Commented Feb 18, 2022 at 19:32