Farey fractions and permutations generated by fractional parts $\{i\alpha\}$

A. V. Shutov

Vladimir State University, Faculty of Physics and Mathematics

Abstract: Let $\alpha\in(0;1)$ be an irrational number. Study of the distribution of fractional parts $\{i\alpha\}$ on the interval $(0;1)$ is a classical question in number theory. In particular, H. Weyl proved that this sequence is uniformity distributed modulo 1. Since this work, various estimates for the remainder term of the asymptotic formula for the number of the sequence of points belonging to a given interval are actively investigated. Another type of problems about considered sequence are problems associated with the famous three lenghts theorem (Steinhaus conjecture), which state that a tiling of the unit interval generated by the points of the sequence, composed of intervals of two or three different lengths. Moreover, in the second case the length of the greatest inteval exactly equals the sum of the lengths of two other intervals. It was find out that the geometry of these tilings is closely connected with the first return maps for circle rotations, Hecke–Kesten problem on bounded remainder sets, combinatorics of Sturmian sequences, dynamics of two-color rotations of the circle, and some other problems.
This paper deals to combinatorial properties of the sequence $\{i\alpha\}$, such as permutations $\pi_{\alpha,n}$, generated by the points $\{i\alpha\}$, $1\leq i\leq n$. It is proved that there is a one-to-one correspondence between these permutations and the intervals of Farey tilings of the level $n$. Here Farey tiling of the level $n$ is a tiling of the interval $ [0;1]$ generated by irreducible rational fractions of the form $ \frac{a}{b}$ with denominator $0<b\leq n$. The proof is based on one theorem of V. T. Sos, that allows to compute the permutation $\pi_{\alpha, n}$ using only $\pi_{\alpha,n}(1)$ and $\pi_{\alpha, n}(n)$. Also we use the fact that the ends of intervals of the Farey coincide with the points of discontinuity of the functions $\{k\alpha\}-\{l\alpha\}$. As an application it is proved that there are exactly $1+\sum_{k = 2}^n \varphi(k)$ different permutations $\pi_{\alpha,n}$ for any fixed $n$. Another our result states that the permutation $\pi_{\alpha,n}$ uniquely determines permutations $\pi_{\alpha,m} $ with $n<m<\pi_{\alpha,n}(1)+\pi_{\alpha, n}(n)$ and does not uniquely determine the permutation $\pi_{\alpha,m} $ with $m=\pi_{\alpha,n}(1)+\pi_{\alpha,n}(n)$.

Keywords: fractional parts, permutations, Farey sequence.

UDC: 519.21

Received: 16.02.2014