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Dilated floor functions having nonnegative commutator. I: Positive and mixed sign dilations. (English) Zbl 1454.11012

The dilated floor function is defined by \(f_\alpha(x)=\lfloor \alpha x\rfloor\) where \(\alpha\in \mathbb{R}\). The compositional commutator of two such functions \([f_\alpha,f_\beta](x)=\lfloor\alpha\lfloor\beta x\rfloor\rfloor- \lfloor\beta\lfloor\alpha x\rfloor\rfloor\) is generally not the zero function. The paper addresses the problem under which conditions we have (*) \(\lfloor\alpha\lfloor\beta x\rfloor\rfloor\ge \lfloor\beta\lfloor\alpha x\rfloor\rfloor\) for all \(x\in \mathbb{R}\). Classification theorem are proved depending on the signs of \(\alpha\) and \(\beta\). Here (*) is true for all \(x\in \mathbb{R}\) in the case of the mixed signs, i.e. when \(\alpha\cdot\beta<0\). From the two remaining cases the simpler case is the positive dilatations classification saying that if \(\alpha>0\) and \(\beta>0\) then (*) is true for all \(x\in \mathbb{R}\) if and only if there are integers \(m,n\ge0\), not both 0, such that \(m\alpha\beta+n\alpha=\beta\). The negative dilatations classification characterization is more technical. The solution set \(S=\{(\alpha,\beta) : [f_\alpha,f_\beta](x)]\ge0 \text{ for all }x\in \mathbb{R}\}\) as a countable union of real semi-algebraic sets having dimensions 2, 1, or 0. The author prove that \(S\) is a closed subset of \(\mathbb{R}^2\). As pointed out by D.Speyer the non-negative commutator property is transitive. A detailed geometric description of the solutions set \(S\) in the positive dilatation case in \((\alpha,\beta)\) coordinates is given. An analogue to Skolem-Bang theorem for disjointness of Beatty sequences for reduced Beatty sequences over \(\mathbb{Z}\) is also proved. (The problem when two dilated floor function commute under composition was solved by the present authors jointly with T. Murayama [Am. Math. Monthly 123, No. 10, 1033–1038 (2016; Zbl 1391.11008)]. In this connection see also [T. Kulhanek et al., PUMP J. Undergrad. Res. 2, 107–117 (2019; Zbl 1439.11017)].

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11B83 Special sequences and polynomials
11D07 The Frobenius problem
11Z05 Miscellaneous applications of number theory
26D07 Inequalities involving other types of functions
52C05 Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
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