Dilated floor functions having nonnegative commutator II. Negative dilations
This paper completes the classification of the set $S$ of all real parameter pairs $(α,β)$ such that the dilated floor functions $f_α(x) = \lfloor{αx}\rfloor$, $f_β(x) = \lfloor{βx}\rfloor$ have a nonnegative commutator, i.e. $ [ f_α, f_β](x) = \lfloor{α\lfloor{βx}\rfloor}\rfloor - \lfloor{β\lfloor{αx}\rfloor}\rfloor \geq 0$ for all real $x$. This paper treats the case where both dilation parameters $α, β$ are negative. This result is equivalent to classifying all positive $α, β$ satisfying $ \lfloor{α\lceil{βx}\rceil}\rfloor - \lfloor{β\lceil{αx}\rceil}\rfloor \geq 0$ for all real $x$. The classification analysis is connected with the theory of Beatty sequences and with the Diophantine Frobenius problem in two generators.
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