Dilated floor functions having nonnegative commutator II. Negative dilations
Volume 196 / 2020
Abstract
This paper completes the classification of the set $S$ of real parameter pairs $(\alpha ,\beta )$ such that the dilated floor functions $\newcommand{\floor}[1]{\lfloor{#1}\rfloor}f_\alpha (x) = \floor {\alpha x}$ and $f_\beta (x) = \floor {\beta x}$ have a nonnegative commutator, i.e. $ [ f_{\alpha }, f_{\beta }](x) = \floor {\alpha \floor {\beta x}} - \floor {\beta \floor {\alpha x}} \geq 0$ for all real $x$. The paper treats the case where both dilation parameters $\alpha , \beta $ are negative. This result is equivalent to classifying all positive $\alpha , \beta $ satisfying $\newcommand{\ceil}[1]{\lceil{#1}\rceil} \floor {\alpha \ceil {\beta x}} - \floor {\beta \ceil {\alpha x}} \geq 0$ for all real $x$. The classification analysis is connected with the theory of Beatty sequences and with the Diophantine Frobenius problem in two generators.
