JACO 939-01 Sturmian Words and the Permutation that Orders {α}, {2α},..., {nα} Kevin O’Bryant [email protected] http://www.math.uiuc.edu/∼obryant February 14, 2002 Abstract We study algebraic properties of the permutation that orders the frac- tional parts {α}, {2α},..., {nα}. Specifically, for each irrational α and positive integer n define πα,n = π to be the permutation of 1, 2, . , n for which 0 < {π(1)α} < {π(2)α} < ··· < {π(n)α} < 1. We give a formula for the sign of πα,n, and also show that for each α, there are infinitely many n for which the multiplicative order of πα,n is less than n. The permutations πα,n have previously been studied combinatorially and analytically, and it is natural to ask whether algebraic properties have any meaning. We demonstrate that these permutations do arise in an algebraic setting. The factors of length n of a Sturmian word can be interpreted as the n + 1 vertices of an n-dimensional simplex. In the investigation of these simplices we are led to define a family of matrices indexed by n and by the slope of the Sturmian word. For a fixed n, we show that these matrices generate a representation of the symmetric group on n symbols, and moreover the matrix formed from a Sturmian word with slope α corresponds to the permutation πα,n. 1 Sturmian words are infinite words over the alphabet {0, 1} that have ex- actly n + 1 factors of length n for each n ≥ 0, e.g., if α is irrational then ∞ (b(n + 1)αc − bnαc)n=1 is Sturmian. They arise in many fields, including game theory, diophantine approximation, and theoretical computer science. Sturmian words are examples of 1-dimensional quasicrystals, at least with respect to some of the ‘working definitions’ currently in use. In contrast to the study of crys- tals, group theory has not been found very useful in the study of quasicrystals. According to M. Senechal [Sen95], “The one-dimensional case suggests that symmetry may be a relatively unimportant feature of aperiodic crystals.” We show that symmetric groups arise naturally when studying the set of all Stur- mian words. Let W be a Sturmian word, and let n T o Fn(W ) := (W (i + 1),W (i + 2),...,W (i + n)) : i ≥ 0 be the set of factors of W of length n, interpreted as column vectors. By the definition of Sturmian, Fn(W ) is a set of n+1 n-dimensional {0, 1}-vectors, and we consider these vectors to be the vertices of a simplex. We show below that 1 the simplex Fn(W ) is not degenerate, and in fact has volume n! , independent of W . We prove this in the following manner. We first translate the simplex Fn(W ) so that a distinguished vertex is at the origin. We then order the non-zero vertices of the translated simplex and list them as columns of a matrix; call this matrix Mn(W ). Surprisingly, the set {Mn(W ): W is Sturmian } generates a faithful representation Σn of Sn, the symmetric group on n symbols. It follows immediately that |det (Mn(W ))| = 1, and consequently vol (Fn(W )) = 1 n! . There is an implied map from {Mn(W ): W is Sturmian} into Sn; we label the image of Mn(W ) under this map πα,n, where |{i ≤ N : W (i) = 1}| α := lim n→∞ N is the slope of W (which must be irrational). Unexpectedly, the permutation πα,n has the property of ordering the fractional parts {α}, {2α},..., {nα}, i.e., with πα,n, 0 < {πα,n(1)α} < {πα,n(2)α} < ··· < {πα,n(n)α} < 1. We are thus led—in our quest to understand the simplex Fn(W ) and the ma- trix Mn(W )—to study the permutation that orders fractional parts of multiples of an irrational. The permutation πα,n has already been studied extensively; we mention two notable results. S´os[S´os57]studied this permutation from a combinatorial view in connection with continued fractions, and gave a recurrence relation giving πα,n(i) in terms of 2 πα,n(i−1), πα,n(1), πα,n(n) and n. An immediate corollary of this recurrence is the famous Three-Distance Theorem, first conjectured by Steinhaus and proven independently by S´os[S´os57], Swierczkowski´ [Swi59],´ Erd˝os,and Sz¨usz. We direct the reader to Slater [Sla67] for historical notes on this problem, and to Alessandri & Berth´e[AB98] for some recent generalizations. Schoißengeier [Sch84] studied this permutation analytically, via Dedekind sums, culminating in his explicit formula for the star-discrepancy of the sequence ∞ ({nα})n=1. To our knowledge, these permutations have not arisen previously in an algebraic setting. We begin a study of algebraic properties of πα,n in Section 3 of this paper. We show that n Y b2`αc sgn(πα,2n) = sgn(πα,2n+1) = (−1) `=1 for all irrational α and all n ≥ 1. We also show, as a corollary of a lemma of S´os[S´os57], that the multiplicative order of π divides φ(n) for infinitely α,n√ 5−1 many n. For some special numbers, such as φ = 2 , it is possible to give more precise statements: ord(πφ,f2n ) = 2 and ord(πφ,f2n+1 ) = 4, where fn is the n-th Fibonacci number. For fixed α, the order of πα,n appears to vary wildly as n → ∞. For example, we have ord(πφ,74) = 79170 and ord(πφ,89) = 4, whereas the expected order of a permutation on 74 (resp. 89) symbols is roughly 10500 (resp. 23700). Our analysis of πα,n leads us back to Sturmian words. In Lemma 3.5(iii) we evaluate the sum (modulo 2) of the Hamming weights of the n + 1 subwords of length n of a Sturmian word. Specifically, X h(~w) ≡ n (mod 2). ~w∈Fn(W ) Note that the right-hand-side of this congruence is independent of W . Section 1 contains a brief—and certainly incomplete—guide to the literature of Sturmian words, Beatty sequences, and the permutation πα,n. In Section 2, we discuss Sturmian words and the representation Σn. In Section 3, we prove some facts about the order and sign of πα,n. Section 4 is a list of questions raised by the results in Sections 2 and 3 that we have been unable to answer. Sections 2 and 3 are logically independent and may be read in any order. A Mathematica notebook containing code for generating the functions and examples in this paper, and many more, is available from the author. 1 A Brief Guide to Related Literature ∞ ∞ A Beatty sequence is a sequence of the form (bnα + βc)n=1 or (dnα + βe)n=1, where α > 0 and β are (possibly rational) real numbers. The Beatty word associated with the Beatty sequence B is a function W : N → {0, 1} such that W (i) = 1 iff i ∈ B. If α is irrational, then the Beatty words associated to ∞ ∞ (bnα + βc)n=1 and (dnα + βe)n=1 are Sturmian, and all Sturmian words arise 3 in this manner. These and many other results basic to the theory of Sturmian words and Beatty sequences are found in [Lot02, Chapter 2]. Several surprising semigroups of matrices have arisen recently in the theory of Beatty sequences. The tip of this iceberg can be seen in the many papers of Stolarsky, Porta, and Fraenkel, in various combinations of authorship, which are listed in the bibliography [FPS90], [Fra94], [FLS72], [PS90], [PS91], [Sto76], [Sto94], [Sto96]. Hankel determinants of Beatty words have been studied in detail, see espe- cially [APWW98] and [KTW99]. In the author’s opinion, the most compelling open problem in the theory of Beatty sequences is Fraenkel’s Conjecture. Fraenkel’s Conjecture: Suppose that m ≥ 3, α1 > α2 > ··· > αm > 1 with strict inequalities, and the m Beatty sequences (bkαi + βic)k≥1 are disjoint and 2m−1 cover the positive integers. Then αi = 2i−1 . 2m−1 i−1 The sequences (bkαi + βic)k≥1 with αi = 2i−1 and βi = 1 − 2 (1 ≤ i ≤ i−1 m) do partition the positive integers, but the choice βi = 1 − 2 is not the only choice that leads to a partition. Fraenkel’s Conjecture has recently been established for m ≤ 6 by Tijdeman [Tij00b]. An up to date (and quite readable) account of Fraenkel’s Conjecture can be found in Tijdeman’s article [Tij00a]. In addition, [Tij00a] contains the most extensive (although by no means complete) bibliography of recent work related to Beatty sequences of which this author is aware. The bibliographies of Brown [Bro93] and Stolarsky [Sto76] cover the early development of the theory. There have been many publications recently concerning Sturmian words and we cannot do justice to this literature. The survey of Berstel [Ber96] gives an excellent overview, and Chapter 2 of the book “Algebraic Combinatorics on Words” [Lot02] gives a rigorous introduction. S´osand Schoißengeier showed that properties of the sequence ({nα})n≥1 are intimately connected to properties of the continued fraction of α. Sto- larsky [Sto76], Brown [Bro93] and others have made explicit the relation be- tween Sturmian words with slope α and the continued fraction expansion of α. One aim of this work is to make a direct connection between Sturmian words and ({nα})-sequences. Boyd & Steele [BS79] give formulas for the length of the longest increasing subsequence of πα,n, based on an explicit solution to a linear programming problem. The works of S´os[S´os57](some of her results are reproved, in English, in [Sla67] and [S´os58])and Schoißengeier have already been mentioned.