Strongly sufficient sets and the distribution of arithmetic sequences in the 3x+13x+1 graph

The 3x+13x+1 conjecture asserts that the TT-orbit of every positive integer contains 11, where TT maps x↦x/2x↦x/2 for xx even and x↦(3x+1)/2x↦(3x+1)/2 for xx odd. A set SS of positive integers is sufficient   if the orbit of each positive integer intersects the orbit of some member of SS. Monks (2006) [8] showed that every infinite arithmetic sequence is sufficient.In this 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/2x↦x/2 and x↦(3x+1)/2x↦(3x+1)/2 modulo bb for bb relatively prime to 66, and study the action of these groups on the directed graph associated to the 3x+13x+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 TT-orbit of every positive integer contains an element congruent to 2mod92mod9, and every non-trivial cycle and divergent orbit contains an element congruent to 20mod2720mod27. We generalize these results to find many other sets that are strongly sufficient in this way.Finally, we show that the 3x+13x+1 digraph exhibits a surprising and beautiful self-duality modulo 2n2n for any nn, 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