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

TL;DR

This paper explores the structure of the 3x+1 problem, constructing sparse sufficient sets, analyzing group actions on the associated graph, and revealing new distributional properties and symmetries in the 3x+1 dynamical system.

Contribution

It introduces new strongly sufficient sets with low density, analyzes group actions modulo various integers, and uncovers a self-duality property of the 3x+1 digraph.

Findings

Every positive integer's orbit contains an element ≡ 2 mod 9.

Non-trivial cycles and divergent orbits contain elements ≡ 20 mod 27.

The 3x+1 digraph exhibits self-duality modulo 2^n.

Abstract

The 3x+1 Conjecture asserts that the T-orbit of every positive integer contains 1, where T maps x\mapsto x/2 for x even and x\mapsto (3x+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. In a previous paper it was shown that every 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\mapsto x/2 and x\mapsto (3x+1)/2 modulo b for b relatively prime to 6, and study the action of these groups on the directed graph associated to the 3x+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. Finally, we show that the 3x+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.