Blurrings of the $j$-function

Inspired by the idea of blurring the exponential function, we define blurred variants of the $j$-function and its derivatives, where blurring is given by the action of a subgroup of $\rm{GL}_2(\mathbb{C})$. For a dense subgroup (in the complex topology) we prove an Existential Closedness theorem which states that all systems of equations in terms of the corresponding blurred $j$ with derivatives have complex solutions, except where there is a functional transcendence reason why they should not. For the $j$-function without derivatives we prove a stronger theorem, namely, Existential Closedness for $j$ blurred by the action of a subgroup which is dense in $\rm{GL}_2^+(\mathbb{R})$, but not necessarily in $\rm{GL}_2(\mathbb{C})$. We also show that for a suitably chosen countable algebraically closed subfield $C \subseteq \mathbb{C}$, the complex field augmented with a predicate for the blurring of the $j$-function by $\rm{GL}_2(C)$ is model theoretically tame, in particular, $\omega$-stable and quasiminimal.