论文标题
Sós排列
Sós Permutations
论文作者
论文摘要
令$ f（x）=αx +β\ mod 1 $ 1 $用于固定实际参数$α$和$β$。对于任何正整数$ n $，将sós排列$π$定义为词典上的第一个排列，以使$ 0 \ leq f（π（0））\ leq f（π（1））\ leq \ cdots \ cdots \ leq f（π（n））<1 $。在本文中，我们在[0,1）^2 $中的参数空间$（α，β）\的分区中进行了sós置换和区域之间的培训。这使我们能够列举这些排列并获得以下“三个区域”定理：在任何垂直条带$（a/b，c/d）\ times [0,1）$中，带有$（a/b，c/d）$ a a farey Interval，最多有三个不同区域的区域，其中一个区域是其他两个区域的总和。
Let $f(x) = αx + β\mod 1$ for fixed real parameters $α$ and $β$. For any positive integer $n$, define the Sós permutation $π$ to be the lexicographically first permutation such that $0 \leq f(π(0)) \leq f(π(1)) \leq \cdots \leq f(π(n)) < 1$. In this article we give a bijection between Sós permutations and regions in a partition of the parameter space $(α,β)\in [0,1)^2$. This allows us to enumerate these permutations and to obtain the following "three areas" theorem: in any vertical strip $(a/b,c/d)\times [0,1)$, with $(a/b,c/d)$ a Farey interval, there are at most three distinct areas of regions, and one of these areas is the sum of the other two.
