Dilated floor functions having nonnegative commutator II. Negative dilations

Jeffrey C. Lagarias, David Harry Richman Acta Arithmetica MSC: Primary 11A25; Secondary 11B83, 11D07, 11Z05, 26D07, 52C05. DOI: 10.4064/aa190628-14-1 Opublikowany online: 3 July 2020

Streszczenie

This paper completes the classification of the set $S$ of real parameter pairs $(\alpha ,\beta )$ such that the dilated floor functions $\newcommand{\floor}[1]{\lfloor{#1}\rfloor}f_\alpha (x) = \floor {\alpha x}$ and $f_\beta (x) = \floor {\beta x}$ have a nonnegative commutator, i.e. $ [ f_{\alpha }, f_{\beta }](x) = \floor {\alpha \floor {\beta x}} - \floor {\beta \floor {\alpha x}} \geq 0$ for all real $x$. The paper treats the case where both dilation parameters $\alpha , \beta $ are negative. This result is equivalent to classifying all positive $\alpha , \beta $ satisfying $\newcommand{\ceil}[1]{\lceil{#1}\rceil} \floor {\alpha \ceil {\beta x}} - \floor {\beta \ceil {\alpha x}} \geq 0$ for all real $x$. The classification analysis is connected with the theory of Beatty sequences and with the Diophantine Frobenius problem in two generators.

Autorzy

  • Jeffrey C. LagariasDepartment of Mathematics
    University of Michigan
    Ann Arbor, MI 48109-1043, U.S.A.
    e-mail
  • David Harry RichmanDepartment of Mathematics
    University of Michigan
    Ann Arbor, MI 48109-1043, U.S.A.
    e-mail

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