The questions around "Collatz" (problem) are questions about the status of the $3n+1$ problem or that of its generalizations and about literature. Since it is an exponential diophantine problem which is iterated according to residues $ \pmod 2$ we deal with modular arithmetic and subsequent problems. Also the general problem of approximation of powers of 2 to that of powers of 3 (in the original $3n+1$-formulation) occurs as determining ingredient.
Since it is an iterative problem, the topic of fixpoints and cyclic points is also relevant.
One might also consider the known strategies of proving; so questions concerned with such strategies of approaching the Collatz-problem occur sometimes.

The Collatz-problem is still unsolved and seemingly very difficult to attack. On the other hand, it is simple to state and understand, so many amateurish approaches exist. Please understand, that MSE (math @ stackexchange) is not intended to check your attempted proofs in general. If you have a question related to something remarkable that you have found, please be specific. For an introductory reading please consult the wikipedia; it has a nice and long overview plus a list of references which you may read online.

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