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

TL;DR

This paper classifies pairs of real dilation parameters for floor functions that have a nonnegative commutator, extending divisibility order and relating to Beatty sequences and Frobenius problems.

Contribution

It provides a classification of dilation pairs with nonnegative commutator, extending divisibility order and connecting to number theory concepts.

Findings

Characterization of parameter pairs with nonnegative commutator

Extension of divisibility order on dilation factors

Connection to Beatty sequences and Frobenius problem

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.