Dilated floor functions having nonnegative commutator II. Negative dilations

@article{Lagarias2019DilatedFF,
  title={Dilated floor functions having nonnegative commutator II. Negative dilations},
  author={Jeffrey C. Lagarias and D. Harry Richman},
  journal={arXiv: Number Theory},
  year={2019}
}
This paper completes the classification of the set $S$ of all real parameter pairs $(\alpha,\beta)$ such that the dilated floor functions $f_\alpha(x) = \lfloor{\alpha x}\rfloor$, $f_\beta(x) = \lfloor{\beta x}\rfloor$ have a nonnegative commutator, i.e. $ [ f_{\alpha}, f_{\beta}](x) = \lfloor{\alpha \lfloor{\beta x}\rfloor}\rfloor - \lfloor{\beta \lfloor{\alpha x}\rfloor}\rfloor \geq 0$ for all real $x$. This paper treats the case where both dilation parameters $\alpha, \beta$ are negative… Expand

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TLDR
All pairs of real numbers (α, β) such that the dilated floor functions commute under composition, i.e., such that ⌊α⌊βx⌋⌌� = ⌋β⊊αx⍋⋋ holds for all real x. Expand
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