The sufficiency of arithmetic progressions for the $3x+1$ Conjecture

Abstract:

Define $T:\mathbb {Z} ^{+}\rightarrow \mathbb {Z} ^{+}$ by $T\left ( x\right ) =\left ( 3x+1\right ) /2$ if $x$ is odd and $T\left ( x\right ) =x/2$ if $x$ is even. The $3x+1$ Conjecture states that the $T$-orbit of every positive integer contains $1$. A set of positive integers is said to be sufficient if the $T$-orbit of every positive integer intersects the $T$-orbit of an element of that set. Thus to prove the $3x+1$ Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets $1+2^{n} \mathbb {N}$ are sufficient for $n\leq 4$ and asked if $1+2^{n}\mathbb {N}$ is also sufficient for larger values of $n$. We answer this question in the affirmative by proving the stronger result that $A+B\mathbb {N}$ is sufficient for any nonnegative integers $A$ and $B$ with $B\neq 0,$ i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.