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arXiv · 2507.14244

Floor, ceiling and the space between

Árpád Bényi·Branko Ćurgus
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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.

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Árpád Bényi, Branko Ćurgus. 2025-07-17. Floor, ceiling and the space between. https://doi.org/10.2478/awutm-2025-0010

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