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arXiv · 1806.00579

Dilated floor functions having nonnegative commutator I. Positive and mixed sign dilations

Jeffrey C. Lagarias·D. Harry Richman
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Abstract

In this paper and its sequel we classify the set $S$ of all real parameter pairs $(α,β)$ such that the dilated floor functions $f_α(x) = \lfloor{αx}\rfloor$ and $f_β(x) = \lfloor{βx}\rfloor$ have a nonnegative commutator, i.e. $ [ f_α, f_β](x) = \lfloor{α\lfloor{βx}\rfloor}\rfloor - \lfloor{β\lfloor{αx}\rfloor}\rfloor \geq 0$ for all real $x$. The relation $[f_α,f_β]\geq 0$ induces a preorder on the set of non-zero dilation factors $α, β$, which extends the divisibility partial order on positive integers. This paper treats the cases where at least one of the dilation parameters $α$ or $β$ is nonnegative. The analysis of the positive dilations case is related to the theory of Beatty sequences and to the Diophantine Frobenius problem in two generators.

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Jeffrey C. Lagarias, D. Harry Richman. 2018-06-19. Dilated floor functions having nonnegative commutator I. Positive and mixed sign dilations. https://doi.org/10.4064/aa180602-21-9

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