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?
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 small volume.
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:
$(1)\ (2)$
$(3)\ (4)$
$(1)$ Single axis translation and discrete scale
$(2)$ Rotation and discrete scale
$(3)$ Scale and discrete rotation
$(4)$ Two axis discrete translation and discrete rotation