The problem can be generalized by considering other functions defined on the natural numbers that are similar to the function T. The functions I consider to be similar to the function T are the periodically linear functions, which are those functions U for which there is a finite modulus d such that the function U when restricted to any congruence class is linear. Some reasons to study generalizations of the problem are that they may uncover new phenomena, they can indicate the limits of validity of known results, and they can lead to simpler, more revealing proofs. Here I discuss three directions of generalizations of the problem. These deal with algorithmic decidability questions, with the existence of stopping times for almost all integers, and with the fractional parts of .