PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Dilated floor functions having nonnegative commutator II. Negative dilations

Volume 196 / 2020

Jeffrey C. Lagarias, David Harry Richman Acta Arithmetica 196 (2020), 163-186 MSC: Primary 11A25; Secondary 11B83, 11D07, 11Z05, 26D07, 52C05. DOI: 10.4064/aa190628-14-1 Published online: 3 July 2020

Abstract

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.

Authors

  • 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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image