Floor, ceiling and the space between

Motivated by questions on the ranges of commutators of dilated floor functions and one posed in Problem 27327 from Gazeta Matematică, we investigate the precise ranges of certain generalized polynomials depending on a real parameter and defined via the floor function. Our analysis requires non-trivial tools, including Kronecker’s approximation theorem. The results highlight sharp distinctions between irrational parameters and sub-unitary and supra-unitary rational parameters. We also propose several conjectures for the irrational and supra-unitary rational cases, supported by extensive computations in Wolfram Mathematica.