@article{Monks2006,
  author    = {{Monks, Kenneth M.}},
  title     = {{The sufficiency of arithmetic progressions for the $3x+1$ conjecture}},
  journal   = {{Proc. Amer. Math. Soc.}},
  year      = {{2006}},
  volume    = {{134}},
  number    = {{10}},
  pages     = {{2861--2872}},
  url       = {{https://monks.scranton.edu/files/pubs/SufficiencyRev4.pdf}},
  abstract  = {{Define $T : \mathbb{Z}^+ \to \mathbb{Z}^+$ by $T(x) = (3x+1)/2$ if $x$ is odd and $T(x) = x/2$ 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 $n \le 4$ 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 analogous results for the Divergent
  Orbits Conjecture and Nontrivial Cycles Conjecture.
}},
  note      = {{MR2231609 (2007c:11030)}},
}