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The 3x+1 Conjecture asserts that the T-orbit of every positive integercontains 1, where T maps x\mapsto x/2 for x even and x\mapsto (3x+1)/2 for xodd. A set S of positive integers is sufficient if the orbit of each positiveinteger intersects the orbit of some member of S. In a previous paper it wasshown that every arithmetic sequence is sufficient.  In this paper we further investigate the concept of sufficiency. We constructsufficient sets of arbitrarily low asymptotic density in the natural numbers.We determine the structure of the groups generated by the maps x\mapsto x/2 andx\mapsto (3x+1)/2 modulo b for b relatively prime to 6, and study the action ofthese groups on the directed graph associated to the 3x+1 dynamical system.From this we obtain information about the distribution of arithmetic sequencesand obtain surprising new results about certain arithmetic sequences. Forexample, we show that the forward T-orbit of every positive integer contains anelement congruent to 2 mod 9, and every non-trivial cycle and divergent orbitcontains an element congruent to 20 mod 27. We generalize these results to findmany other sets that are strongly sufficient in this way.  Finally, we show that the 3x+1 digraph exhibits a surprising and beautifulself-duality modulo 2^n for any n, and prove that it does not have thisproperty for any other modulus. We then use deeper previous results toconstruct additional families of nontrivial strongly sufficient sets by showingthat for any k<n, one can "fold" the digraph modulo 2^n onto the digraph modulo2^k in a natural way.

Strongly sufficient sets and the distribution of arithmetic sequences in the 3x+1 graph