具有非负换位子II的扩张地板函数。负性扩张
Dilated floor functions having nonnegative commutator II. Negative dilations
摘要：
本文完成了所有实参数对$(α,β)$的集合$S$的分类，使得扩张的地板函数$f_α(x) = \lfloor{αx}\rfloor$，$f_β(x) = \lfloor{βx}\rfloor$有一个非负的交换子，即对所有实$x$有$ [ f_α, f_β](x) = \lfloor{α\lfloor{βx}\rfloor}\rfloor - \lfloor{β\lfloor{αx}\rfloor}\rfloor \geq 0$.本文处理的情况下，两个膨胀参数$α, β$是负的。该结果等同于对所有真实$x$的满足$ \lfloor{α\lceil{βx}\rceil}\rfloor - \lfloor{β\lceil{αx}\rceil}\rfloor \geq 0$的所有阳性$α, β$进行分类。分类分析与Beatty序列的理论和Diophantine Frobenius问题在两个生成器。
英文摘要：
This paper completes the classification of the set $S$ of all real parameter pairs $(α,β)$ such that the dilated floor functions $f_α(x) = \lfloor{αx}\rfloor$, $f_β(x) = \lfloor{βx}\rfloor$ 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 $x$. This paper treats the case where both dilation parameters $α, β$ are negative. This result is equivalent to classifying all positive $α, β$ satisfying $ \lfloor{α\lceil{βx}\rceil}\rfloor - \lfloor{β\lceil{αx}\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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
