Dilated Floor Functions That Commute Sometimes

Abstract

We explore the dilated floor function f_a(x)= ⌊ax⌋ and its commutativity with functions of the same form. A previous paper found all a and b such that f_a and f_b commute for all real x. In this paper, we determine all x for which the functions commute for a particular choice of a and b. We calculate the proportion of the number line on which the functions commute. We determine bounds for how far away the functions can get from commuting. We solve this fully for integer a, b and partially for real a, b.