Dilated floor functions having nonnegative commutator I. Positive and mixed sign dilations

Abstract

In this paper and its sequel we classify the set $S$ of all real parameter pairs $(\alpha,\beta)$ such that the dilated floor functions $f_\alpha(x) = \lfloor{\alpha x}\rfloor$ and $f_\beta(x) = \lfloor{\beta x}\rfloor$ 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 $x$. The relation $[f_\alpha,f_\beta]\geq 0$ induces a preorder on the set of non-zero dilation factors $\alpha, \beta$, which extends the divisibility partial order on positive integers. This paper treats the cases where at least one of the dilation parameters $\alpha$ or $\beta$ is nonnegative. The analysis of the positive dilations case is related to the theory of Beatty sequences and to the Diophantine Frobenius problem in two generators.