Sós Permutations

Abstract

Let $f(x)=\alpha x+\beta \mathrm{mod}1$ for fixed real parameters $\alpha$ and $\beta$. For any positive integer $n$, define the Sós permutation $\pi$ to be the lexicographically first permutation such that $0\le f(\pi (0))\le f(\pi (1))\le \cdots \le f(\pi (n))<1$. In this article we give a bijection between Sós permutations and regions in a partition of the parameter space $(\alpha ,\beta )\in [0,1{)}^2$. 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.

15 pages