Floor, ceiling and the space between

TL;DR

This paper investigates the ranges of generalized polynomials involving floor functions, revealing sharp differences based on the nature of the parameters, and employs advanced tools like Kronecker's theorem to analyze these distinctions.

Contribution

It provides a detailed analysis of the ranges of certain floor-based polynomials depending on parameter types, introducing new conjectures and computational evidence.

Findings

Distinct behaviors for irrational and rational parameters

Sharp range distinctions between sub-unitary and supra-unitary rationals

Proposed conjectures supported by computational experiments

Abstract

Motivated by questions on the ranges of commutators of dilated floor functions and one posed in Problem 27327 from Gazeta Matematic\u{a}, 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.