Dilated floor functions having nonnegative commutator II. Negative dilations

Jeffrey C. Lagarias, D. Harry Richman

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_β(x) = \lfloor{βx…