Sós Permutations

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.

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