Let $A$ be a finite dimensional quiver algebra and $M$ a finite dimensional $A$-module. Assume we want to test whether $M$ is a periodic module, meaning that $\Omega^n(M) \cong M$ for some $n \geq 1$. The smallest such $n$ is then called the period of the module. Assume that $A$ has the property that a module is periodic iff it is bounded (meaning that the dimensions of $\Omega^k(M)$ have a finite bound). This condition was first investigated (I think) by Rainer Schulz, see https://www.sciencedirect.com/science/article/pii/0021869386902048 . I think this condition is for example satisfied for all group algebras and I think there are no known examples over finite fields where this is not true. It can be tested with the computer whether a given module has period at most $k$ when one enters $M$ and a number $k$ (for example using the GAP-package QPA). But there is the problem: In case $M$ is periodic, the program eventually stops but if $M$ is not periodic the computer cant say for sure and might calculate forever.

This motivated the following questions:

Is there a useful "bound" s(A,M) that just depends on the vector space dimension of $A$ and $M$ such that $M$ is not periodic in case there exists a $k$ such that $\Omega^k(M)$ has dimension at least $s(A,M)$?

In case such a theoretical result would be available, testing whether a module is periodic would be a finite problem.

I have not much experience about that, but I would guess that $s(A,M)=2 dim(A) dim(M) +30$ might do?