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The 3x+1 Conjecture asserts that the T-orbit of every positive integer 1, where T maps x\ x/2 for x even and x\ (3x+1) /2 for x. A set S of positive integers is sufficient if the orbit of each positive intersects the orbit of some member of S. In a previous paper it was that every arithmetic sequence is sufficient. In this paper we further investigate the concept of sufficiency. We construct sets of arbitrarily low asymptotic density in the natural numbers. determine the structure of the groups generated by the maps x\ x/2 and\ (3x+1) /2 modulo b for b relatively prime to 6, and study the action of groups on the directed graph associated to the 3x+1 dynamical system. this we obtain information about the distribution of arithmetic sequences obtain surprising new results about certain arithmetic sequences. For, we show that the forward T-orbit of every positive integer contains an congruent to 2 mod 9, and every non-trivial cycle and divergent orbit an element congruent to 20 mod 27. We generalize these results to find other sets that are strongly sufficient in this way. Finally, we show that the 3x+1 digraph exhibits a surprising and beautiful-duality modulo 2ⁿ for any n, and prove that it does not have this for any other modulus. We then use deeper previous results to additional families of nontrivial strongly sufficient sets by showing for any k<n, one can "fold" the digraph modulo 2ⁿ onto the digraph modulo2ᵏ in a natural way.
Monks et al. (Tue,) studied this question.