Dilated floor functions having nonnegative commutator II. Negative dilations
| Autor: | Lagarias, Jeffrey C., Richman, D. Harry |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Acta Arithmetica 196 (2020), no. 2, 163--186 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.4064/aa190628-14-1 |
| Popis: | This paper completes the classification of the set $S$ of all real parameter pairs $(\alpha,\beta)$ such that the dilated floor functions $f_\alpha(x) = \lfloor{\alpha x}\rfloor$, $f_\beta(x) = \lfloor{\beta x}\rfloor$ have a nonnegative commutator, i.e. $ [ f_{\alpha}, f_{\beta}](x) = \lfloor{\alpha \lfloor{\beta x}\rfloor}\rfloor - \lfloor{\beta \lfloor{\alpha x}\rfloor}\rfloor \geq 0$ for all real $x$. This paper treats the case where both dilation parameters $\alpha, \beta$ are negative. This result is equivalent to classifying all positive $\alpha, \beta$ satisfying $ \lfloor{\alpha \lceil{\beta x}\rceil}\rfloor - \lfloor{\beta \lceil{\alpha x}\rceil}\rfloor \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. Comment: 18 pages, 8 figures |
| Databáze: | arXiv |
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