Dilated Floor Functions That Commute

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 $x$.

6 pages, to appear in Amer. Math. Monthly