@article{KrasikovLagarias2003,
  author    = {{Ilia Krasikov and J. C. Lagarias}},
  title     = {{Bounds for the $3x+1$ problem using difference inequalities}},
  journal   = {{Acta Arithmetica}},
  year      = {{2003}},
  volume    = {{109}},
  number    = {{3}},
  pages     = {{237--258}},
  url       = {{https://eudml.org/doc/278462}},
  abstract  = {{We study difference inequality systems for the 3x+1 problem introduced by the first author in 1989. These systemes can be used to give lower bounds for the number of integers below x that contain 1 in their forward orbit under the 3x+1 map. Previous methods gave away some information in these inequalities. We give an improvement which apparantly extracts full information from the inequalities. By computer aided proof we show that at least x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map.}},
  note      = {{MR 2004i:11020}},
  extra_urls= {{https://arxiv.org/abs/math/0205002}},
}