Dilated floor functions having nonnegative commutator II. Negative dilations

Abstract

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. $[f_{\alpha },f_{\beta }](x)=\lfloor \alpha \lfloor \beta x\rfloor \rfloor -\lfloor \beta \lfloor \alpha x\rfloor \rfloor \ge 0$ for all real $x$. This paper treats the case where both dilation parameters $\alpha ,\beta$ are negative. This result is equivalent to classifying all positive $\alpha ,\beta$ satisfying $\lfloor \alpha \lceil \beta x\rceil \rfloor -\lfloor \beta \lceil \alpha x\rceil \rfloor \ge 0$ for all real $x$. The classification analysis is connected with the theory of Beatty sequences and with the Diophantine Frobenius problem in two generators.

arXiv comment18 pages, 8 figures

Cited by in corpus (1)

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