Strongly sufficient sets and the distribution of arithmetic sequences in the graph

The 3 x + 1 conjecture asserts that the T -orbit of every positive integer contains 1 , where T maps x ↦ x / 2 for x even and x ↦ ( 3 x + 1 ) / 2 for x odd. A set S of positive integers is sufficient if the orbit of each positive integer intersects the orbit of some member of S . Monks (2006) [8] showed that every infinite arithmetic sequence is sufficient. s paper we further investigate the concept of sufficiency. We construct sufficient sets of arbitrarily low asymptotic density in the natural numbers. We determine the structure of the groups generated by the maps x ↦ x / 2 and x ↦ ( 3 x + 1 ) / 2 modulo b for b relatively prime to 6 , and study the action of these groups on the directed graph associated to the 3 x + 1 dynamical system. From this we obtain information about the distribution of arithmetic sequences and obtain surprising new results about certain arithmetic sequences. For example, we show that the forward T -orbit of every positive integer contains an element congruent to 2 mod 9 , and every non-trivial cycle and divergent orbit contains an element congruent to 20 mod 27 . We generalize these results to find many other sets that are strongly sufficient in this way. y, we show that the 3 x + 1 digraph exhibits a surprising and beautiful self-duality modulo 2 n for any n , and prove that it does not have this property for any other modulus. We then use deeper previous results to construct additional families of nontrivial strongly sufficient sets by showing that for any k < n , one can “fold” the digraph modulo 2 n onto the digraph modulo 2 k in a natural way.