I posted this is mathematics SE and it didn't get much attention, so I'm trying here. I apologize if it is not relevant.


Take a set $X \in \mathbb{R}^2$ of nonzero measure $\mu(X) \neq 0$. I am attempting to design a set that has the following symmetries (continuous or discrete)


$1.$ Scale symmetry


$2.$ Rotation symmetry


$3.$ Translation symmetry


It's intuitively clear that only $X = \mathbb{R}^n$ itself simultaneously satisfies those 3 symmetries (which I call the trivial solution).


It seems that when you get one symmetry, you are forced to make the second discrete, and you cannot take a non-trivial third symmetry.


Is there a reference that would help me in my task? Is it impossible in some sense that I should be aware of?




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The application I have in mind is recognizing an 'Engineered pattern', that is, the set $X$ within a 2D image.


To search a pattern we need to search the space of rotations (parametrized by $\theta$), translations (parametrized by $\Delta\_x, \Delta\_y$) and scaling (parametrized by $S$), that is, a cube like $[0,360]\times[0,720]\times[0,1280]\times[1/5,5]$, which is quite extensive to search into due to the high dimensionality.


By making the pattern symmetric I hope I can instead restrict my search to a cube of smaller volume: if the pattern is symmetric w.r.t. parameters $P\_1, ..., P\_k$, it is enough to search the space of parameters $P\_{k+1}, ..., P\_n$, which has reduced volume.




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In practice the pattern will have some noise superposed and the rest a picture alongside. The method I was aiming for searching is finding maxima of the cross correlation of the pattern with the image.


Here are some sketches of patterns that illustrate the issue of conflicting symmetries:


<https://i.stack.imgur.com/xskYo.png>


$(1)\ (2)$


$(3)\ (4)$


$(1)$ Single axis translation and discrete scale


$(2)$ Rotation and discrete scale 


$(3)$ Scale and discrete rotation


$(4)$ Two axis discrete translation and discrete rotation



Tradeoff among symmetries
It is ok if your set $X$ is closed, and your rotation is different from a half-turn.


The first example in your picture is invariant by the half-turn around the origin, so you need more than just an half-turn.


A non-closed counterexample would be the complement of ${\bf Q}^2$. This set is invariant by a translation by a rational vector, a homothety with rational scaling factor, and some rotation such that the cosine and sine of the angle are rational (e.g. $\arccos(3/5)$), just because ${\bf Q}^2$ is invariant by these transformations.


If you don't want to assume $X$ to be closed, it is still possible to show that $X$ has a complement of zero measure.


Here is a sketch of proof. The set $X$ is invariant by a translation, and also by another translation in different direction, obtained by conjugating the first translation by the rotation. Choose a frame of ${\bf R}^2$ along these two directions, such that the origin belongs to $X$ and is a Lebesgue density point. Restricting our attention to one of the axis of our frame and using the scaling symmetry, it is not difficult to show that there is a dense subset of the axis that belongs to $X$. Same for the other axis. Because the origin is a density point, 
there is a small rectangle $R$ around the origin in $X$ such that $R\cap X$ spawns a large fraction of $R$. Now scaling and translating back close to the origin, we get an arbitrary large rectangle $R'$ near the origin such that $R'\cap X$ spawns a large fraction of $R'$, which is sufficient to conclude.


