Dilated Floor Functions That Commute

TL;DR

This paper characterizes all pairs of real numbers for which the dilated floor functions commute under composition, providing a complete solution to the functional equation involving these functions.

Contribution

It completely determines the conditions on  and eta for the dilated floor functions to commute, solving a specific functional equation.

Findings

Identifies all commuting pairs (, eta) for dilated floor functions.

Provides a complete characterization of the functional equation solutions.

Advances understanding of the algebraic structure of floor functions.

Abstract

We determine all pairs of real numbers $(\alpha, \beta)$ such that the dilated floor functions $\lfloor \alpha x\rfloor$ and $\lfloor \beta x\rfloor$ commute under composition, i.e., such that $\lfloor \alpha \lfloor \beta x\rfloor\rfloor = \lfloor \beta \lfloor \alpha x\rfloor\rfloor$ holds for all real $x$.