Dilated Floor Functions That Commute Sometimes - Kulhanek, Travis; McDonough, Joseph; Ponomarenko, Vadim (The PUMP Journal of Undergraduate Research, 2019)
We explore the dilated floor function fa(x)= ⌊ax⌋ and its commutativity with functions of the same form. A previous paper found all a and b such that fa and fb 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.