POWERS OF RATIONALS MODULO 1 AND RATIONAL BASE NUMBER SYSTEMS

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1 ISRAEL JOURNAL OF MATHEMATICS 68 (8), 53 9 DOI:.7/s POWERS OF RATIONALS MODULO AND RATIONAL BASE NUMBER SYSTEMS BY Shigeki Akiyama Deartment of Mathematics, Niigata University, Jaan akiyama@math.sc.niigata-u.ac.j AND Christiane Frougny LIAFA, UMR 789 CNRS and Université Paris 8, France Christiane.Frougny@liafa.jussieu.fr AND Jacues Sakarovitch LTCI, UMR 54, CNRS / ENST, 46, rue Barrault, Paris Cedex 3, France sakarovitch@enst.fr ABSTRACT A new method for reresenting ositive integers and real numbers in a rational base is considered. It amounts to comuting the digits from right to left, least significant first. Every integer has a uniue exansion. The set of exansions of the integers is not a regular language but nevertheless addition can be erformed by a letter-to-letter finite right transducer. Every real number has at least one such exansion and a countable infinite number of them have more than one. We exlain how these exansions can be aroximated and characterize the exansions of reals that have two exansions. The results that we derive are ertinent on their own and also as they relate to other roblems in combinatorics and number theory. A first examle is a new interretation and exansion of the constant K() from the so-called Josehus roblem. More imortant, these exansions in the base allow us to make some rogress in the roblem of the distribution of the fractional art of the owers of rational numbers. Work artially suorted by the CNRS/JSPS contract 3 569, and by the ACI Nouvelles Interfaces des Mathématiues, contract 4 3. Aril 7, 6 and in revised form January 7, 7 53

2 54 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math.. Introduction The distribution modulo of the owers of a rational number, indeed the roblem of roving whether they form a dense set or not, is a frustrating uestion: This very old roblem of Pisot and Vijayaraghavan is still unanswered writes Michel Mendés France in [6] and he goes on: Pisot, Vijayaraghavan and André Weil did however show that there are infinitely many limit oints (cf. [5], for instance.) With this roblem as a background, Mahler asked in [5] whether there exists a nonzero real z such that the fractional art of z(3/) n for n =,,... fall into [, /[. It is not known whether such a real, called Z-number, does exist but Mahler showed that the set of Z-numbers is at most countable. His roof is based on the fact that the fractional art of a Z-number (if it exists) has an exansion in base 3/ which is entirely determined by its integral art. In this aer, we introduce and study a new method for reresenting ositive integers and real numbers in the base, where > are corime integers. While this new method does not solve Mahler s original roblem, it sheds a new light on the uestion and allows us to make some rogress on the commonly studied generalization of Mahler s roblem as we exlain at the end of this introduction. The idea of nonstandard reresentation systems of numbers is far from being original and there have been extensive studies of these, from a theoretical standoint as well as for imroving comutation algorithms. It is worth (briefly) recalling first the main features of these systems in order to clearly ut in ersective and in contrast the results we have obtained on rational base systems. Many nonstandard numeration systems have been considered in the literature: [3, Vol., Cha. 4] or [4, Cha. 7], for instance, give extensive references. Reresentation in integer base with signed digits was oularized in comuter arithmetic by Avizienis [] and can even be found earlier in a work of Cauchy [4]. When the base is a noninteger real number β >, any nonnegative real number is given an exansion on the canonical alhabet {,,..., β } by the greedy algorithm of Rényi []; a number may have several β-reresentations on the canonical alhabet, but the greedy one is the greatest in the lexicograhical order. The set of greedy β-exansions of numbers of [, [ is shift-invariant, and

3 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 55 its closure forms a symbolic dynamical system called the β-shift. The roerties of the β-shift are well-understood, using the so-called β-exansion of, see [8, 4]. When β is a Pisot number, the β number system shares many roerties with the integer base case: the set of greedy reresentations is recognizable by a finite automaton; the conversion between two alhabets of digits (in articular addition) is realized by a finite transducer []. In this work, we first define the -exansion of an integer N: it is a way of writing N in the base by an algorithm which roduces least significant digits first. We rove Theorem : Every non-negative integer N has a -exansion which is an integer reresentation. It is the uniue finite -reresentation of N. The -exansions are not the -reresentations that would be obtained by the classical greedy algorithm in base. They are written on the alhabet A = {,,..., }, and not every word of A is admissible. These -exansions share some roerties with the exansions in an integer base digit set conversion is realized by a finite automaton, for instance and are comletely different as far as other asects are concerned. Above all, the set L of all -exansions is not a regular language (not even a context-free one). By construction, the set L is refix-closed and any element can be extended (to the right) in L. Hence, L is the set of labels of the finite aths in an infinite subtree T of the infinite full -ary tree of the free monoid A. The tree T contains a maximal infinite word t maximal in the lexicograhic ordering whose numerical value is ω. We consider the set of infinite words W, subset of A N, that label the infinite aths of T as the admissible -exansions of real numbers and we rove Theorem : Every real in [, ω ] has exactly one -exansion, but for an infinite countable subset of reals which have more than one such exansion. If then no real has more than two -exansions. It is noteworthy as well that no -exansion is eventually eriodic and thus in articular and in contrast with the exansion of reals in an integer base no -exansion ends with ω or, which is the same, is finite. This is a very remarkable feature An algebraic integer > whose Galois conjugates are all less than in modulus.

4 56 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. of the number system for reals and we exlain how the -exansion of a real number can be comuted (in fact, aroximated). We shall give here two examles of the relations of the -exansions of reals with other roblems in combinatorics and number theory. The first one is the so-called Josehus roblem in which a certain constant K() is defined (cf. [7,, 4]) which is a secial case of our constant ω (with = ) and this definition yields a new method for comuting K(). The connection with the second roblem, namely, the distribution of the owers of a rational number modulo with which we oened this introduction, is even more striking. In order to describe this connection, let us first set the framework of this deely intriguing roblem. Koksma roved that for almost every real number θ > the seuence {θ n } is uniformely distributed in [, ], but very few results are known for secific values of θ. One of these is that if θ is a Pisot number, then the above seuence converges to if we identify [, [ with R/Z. Exerimental results show that the distribution of {( n } ) for corime ositive integers > looks more chaotic than the distribution of the fractional art of the owers of a transcendental number like e or π (cf. [6]). The next ste in attacking this roblem has been to fix the rational and to study the distribution of the seuence f n (z) = { z ( ) n } according to the value of the Once again, the seuence f n (z) is uniformly distributed for almost all z >, but nothing is known for secific values of z. In the search for z s for which the seuence f n (z) is not uniformly distributed, and as already exlained, Mahler considered those for which the seuence is eventually contained in [, [. Mahler s notation is generalized as follow: let I be a (strict) subset of [, [ indeed I will be a finite union of semi-closed intervals and write Z ( I ) = { z R : { ( ) } n z } stays eventually in I. ([ Mahler s roblem is to ask whether Z3, [) is emty or not. This resentation is based on the introduction of [3]. The fractional art of a number x is denoted by {x}.

5 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 57 Mahler s work has been develoed in two directions: the search for subsets I as large as ossible such that Z ( I ) is emty and conversely the search for ( ) subsets I as small as ossible such that Z I is nonemty. Along the first line, remarkable rogress has been made by Flatto et al. ([8]) ([ who roved that the set of reals s such that Z s, s + [) is emty, is dense in [, ]. Recently Bugeaud [3] roved that its comlement is of Lebesgue ([ measure. Along the other line, Pollington [] showed that 4 Z3 65, 65[) 6 is nonemty. Our contribution to the roblem can be seen as an imrovement of this last result. Theorem 3: If, there exists a subset Y measure, such that Z ( ) Y is countably infinite. The elements of Z ( Y of [, [, of Lebesgue ) are indeed the reals which have two -exansions (cf. Theorem 49) and this is the reason why the consideration of the number system allowed to make some rogress in Mahler s roblem. Coming back to the historical 3/ case, we have Corollary 4: The set of ositive numbers z such that { z is countably infinite. ( ) n } 3 [, /3[ [/3, [ for n =,,,... It is noteworthy that the exansion comuted by Mahler for his Z-numbers haens to be exactly one of our 3 -exansions if it exists. Another way to state Corollary 4 is the following. Let us denote by x the distance between x and the closest integer. Corollary 4 assures that there are (infinitely many) ositive numbers x such that x(3/) n < /3 for n =,,.... This is to be comared with a recent result of Dubickas [6] who showed that x(3/) n < (n =,,...) imlies that x = hence extending his result [5] on the distribution of {xα n } which works basically for any algebraic number α. Though there is a distance between /3 and , we exect that our Y is minimal in the sense that for any roer subset X of Y which is a finite ( ) union of half oen intervals, Z X is emty, an even stronger statement than Mahler s conjecture (cf. (7) at the very end of this aer). Further study on

6 58 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. the connection between Mahler s roblem and number system is carried out in []. We have introduced and studied here a fascinating object which can be seen from many sides, which raises many difficult uestions and whose further study will certainly mix techniues from word combinatorics, automata theory, and number theory.. Preliminaries.. Finite and infinite words. An alhabet A is a finite set. Here we consider alhabets of digits, that is, subsets of the integers. A finite seuence of elements of A is called a word, and the set of words on A, euied with concatenation is the free monoid A. The emty word, denoted by ε, is the identity element of A. The length of a word w is eual to the length of the seuence w and is denoted by w. The set of words on A of length n (res., of length smaller than or eual to n) is denoted by A n (res., A n ); the concatenation of w reeated n times is denoted by w n. A word u is a factor of a word w if there exist words x and y such that w = x u y. If x (res., y) is the emty word, then u is a refix (res., a suffix) of w. A subset of A is refix-closed (res., suffix-closed) if it contains all refixes (res., all suffixes) of any of its elements. Let us suose that A is ordered by a total order written, which is rather natural as our alhabets are subsets of N or Z. The set A is totally ordered by the radix order defined as follows 3 : v w if v < w, or v = w and there exist letters a < b such that v = u a v and w = u b w. The set A is also totally ordered by the lexicograhic order defined as follows: v w if v is a refix of w, or there exist letters a < b such that v = u a v and w = u b w. The radix order is a well order whereas the lexicograhic order is not (if A has more than one letter). Both orders coincide for air of words of eual length. An infinite word over A is an infinite seuence of elements of A. In this work, infinite words can be indexed by ositive, or negative, integers, deending on the context; in both cases, we denote by A N the set of infinite words on A 3 It is easier to describe the nonreflexive art of the order.

7 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 59 and whenever it is ossible we denote infinite words by bold letters. The refix of length n of a is denoted a [n], but its n-th letter is more lightly written a n. The lexicograhic order is defined on A N as follows: a b if there exist letters a < b such that a = u a a and b = u b b. An infinite word is said to be eventually eriodic if it is of the form u v ω = u v v v v v where u and v belong to A. The set A N is euied with the distance δ defined by: if a = (a n ) n N and b = (b n ) n N, then δ(a,b) = r if a b and r = min{n : a n b n }, and δ(a,b) = if a = b. The toology on the set A N is then the roduct toology (of the discrete toology on A), and it makes A N a comact metric sace... Automata and transducers. An automaton on A, A= Q, A, E, I, T, is a labeled grah: Q is the set of vertices, traditionally called states; I and T are two subsets of Q, the sets of initial and final states resectively; and E, the set of edges, traditionally called transitions, labeled in A, is (or can be seen as) a subset of Q A Q. The transosed of A is the automaton A t = Q, A, E t, T, I where (, a, ) is in E t if, and only if, (, a, ) is in E. The automaton A is deterministic if it has only one initial state, and if for every air (, a) in Q A, there exists at most one such that (, a, ) is in E; the automaton A is co-deterministic if its transosed is deterministic. A state is accessible if there exists a ath from an initial state to, the accessible art of A is the subgrah induced by the set of accessible states. A successful ath in A is a ath whose origin is in I and its end in T. A word in A is acceted by A if it is the label of a successful ath. Figure (a) shows how automata are deicted; in articular, initial states are marked with incoming arrows and final states with outgoing arrows. An automaton (on a finite alhabet) is finite if it has a finite set of states. A language on A, that is, a subset of A, is regular if it is the set of words acceted by a finite automaton on A. Indeed, we shall consider automata whose transitions are labeled in A B where B is another alhabet rather than in A, and which we call transducers. Pairs of words are multilied comonent wise, that is, A B is a monoid, and the label of a (successful) ath in such a transducer is a air of words. A transducer realizes then a relation from A into B. Figure (b) shows a transducer Q that realizes the integral division by 3 on binary reresentations of numbers: (f, g) is acceted by Q if f is the binary reresentation of

8 6 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. (a) An automaton for the numbers divisible by 3 (, ) (, ) (, ) (, ) (, ) (, ) (b) A transducer for the uotient by 3 Figure. An automaton on {, } and another one on {, } {, } that read, and write, numbers written in the binary system. an n divisible by 3 and g is the binary reresentation of n/3, ossibly refixed with some s. For more definitions and results on automata theory the reader is referred to [7], [], or [], to uote a few. Three more things should be added though. First, the transitions of the transducers we shall consider are labeled in A B we call these transducers letter-to-letter. If one retains the first comonent of the labels of a letter-to-letter transducer T one gets an automaton (on A): the underlying inut automaton of T. A transducer is seuential (res., co-seuential) if its underlying inut automaton is deterministic (res., codeterministic). Second, we shall consider automata where the outgoing arrows are labeled, with airs of the form (ε, h); this means that if a ath in A from i in I to t in T is labeled with (f, g) and if the outgoing arrow from t is labeled with (ε, h), then f is associated with g h by the relation realized by A. Finally, the label of a ath has been imlicitly understood as the concatenation from left to right of the label of transitions that constitute the ath. But one could consider automata which read (and write) words from right to left; we call them right automata, or right transducers. An examle of a transducer with these two further characteristics is the one shown at Figure that

9 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 6 realizes the addition in the binary system or, more recisely, the conversion of reresentations written in the digit alhabet {,, } into reresentations written in the classical binary alhabet {, }. (, ) (, ) (, ) (, ) (, ) (ε, ) (, ) Figure. The converter from {,, } in the binary system..3. Reresentation of numbers. Let U = {u i : i Z} be a strictly increasing seuence of ositive real numbers such that for any k, k i u i < +. A reresentation in the system U of a nonnegative real number x on a finite alhabet of digits D is an infinite seuence (d i ) k i, with k in Z and every d i in D, such that i=k x = d i u i. It is denoted by x U = d k d.d d, most significant digit first. When a reresentation ends in infinitely many zeroes, it is said to be finite, and the trailing zeroes are omitted. When all the d i at the right of the radix oint are zeroes, the reresentation is said to be an integer reresentation. Conversely, the numerical value in the system U of a word on an alhabet of digits D is given by the evaluation ma π: i=k π: D Z R, d = (d i ) k i π(d) = d i u i. 3. Reresentation of the integers 3.. The Modified Division algorithm and the number system. Let > be two co-rime integers. Let N be any ositive integer; let us write

10 6 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. N = N and, for i, write () N i = N i+ + a i where a i is the remainder of the division of N i by, and thus belongs to A = {,..., }. Since N i+ is strictly smaller than N i, the division () can be reeated only a finite number of times, until eventually N k+ = for some k. The seuence of successive divisions () for i = to i = k is thus an algorithm that in the seuel is referred to as the Modified Division, or MD algorithm which, given N, roduces the digits a, a,..., a k, and it holds () N = k i= a i ( ) i. We will say that the word a k a, comuted from N from right to left, that is to say least significant digit first, is a -reresentation of N. Since we will show that this reresentation is uniue in Theorem, it will be called the -exansion of N and written. By convention, the N -exansion of is the emty word ε. Examle : Let = 3 and =, then A = {,, } this will be our main running examle. Table gives the 3 -exansions of the eleven first non negative integers. ε Table. Following the notations of Section.3, let U be the seuence defined by: U = {u i = ( ) i : i Z}.

11 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 63 We will say that U, together with the alhabet A = {,..., }, is the number system. If =, it is exactly the classical number system in base. It is to be stressed that this definition is not the classical one, the so-called beta-exansions, see [] and [4, Chater 7], for the numeration system in base : U is not the seuence of owers of but rather these owers divided by and the digits are not the integers smaller than but rather the integers whose uotient by is smaller than. If, on the contrary, = the MD algorithm gives the same exansion as the one given by the classical greedy algorithm, since the exansion is uniue. As stated in the following lemma, one of the main roerties of the classical integer base system is nevertheless retained. Lemma 5: Let π: A Q be the evaluation ma associated with the number system. The restriction of π to A k, for any k, is injective. Proof. Let u = a k a k a and v = b k b k b be two words of A of length k such that π(u) = π(v). Hence k ( ) i k ( ) i a i b i = i= i= and therefore k i= (a i b i )X i is a olynomial in Z[X] vanishing at X =. By Gauss Lemma on rimitive olynomials, it is then divisible by the minimal olynomial X. Contradiction, since the absolute value of the constant term a b is strictly smaller than. It is not true that π is injective on the whole A since for any u in A and any integer h it holds that: π( h u) = π(u). On the other hand, Lemma 5 imlies that this is the only ossibility and we have: (3) π(u) = π(v) and u > v = u = h v with h = u v. Theorem : Every nonnegative integer N has a -exansion which is an integer reresentation. It is the uniue finite -reresentation of N. Proof. Let a k a be the -exansion given to N by the MD algorithm, and suose that there exists another finite reresentation of N in the system U,

12 64 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. of the form e l e l e.e e m with e m. Then ( ) m l ( ) m+i k ( N = e i = a i i= m i= ) m+i and therefore π(e l e e e e m ) = π(a k a k a m ). Contradiction between (3) and e m. Thus the word a k a of A is the uniue finite -reresentation of N (with the condition that a k ) and we denote N = a k a. 3.. The set of -exansions of the integers. Let us denote by L the set of -exansions of the nonnegative integers. If =, then L is the set of all words of A which do not begin with a ; if we release this last condition, we then get the whole A Right contexts. By construction, L is refix-closed; the observation of Table shows that it is not suffix-closed if. In the seuel we assume that, unless it is stated otherwise. Let n and k be natural integers and let us denote by RC k (n) the set of words of length smaller than k + that can be suffixed to the -exansion of n and still form words of L : RC k (n) = {w A k : n w L }. Lemma 6: Let n and m be two nonnegative integers. A word w of length k belongs to both RC k (n) and RC k (m) if and only if n and m are congruent modulo k and in this case RC k (n) = RC k (m). That is: { } n w L and m w L n m (mod k ) RC k (n) = RC k (m). Proof. The word n w belongs to L if and only if ( )k n + π(w) is in N, and similarly for m. Thus: { } ( ) k n w L and m w L (n m) Z n m (mod k ) since and are corime. Conversely, suose that n m (mod k ) then n m (mod h ) for any h k. Hence for every word w of length h k such that ( )h n + π(w) is in N, so is ( )k m + π(w), and RC k (n) = RC k (m).

13 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 65 Lemma 6 imlies immediately that the coarsest right regular euivalence that saturates L is the identity, hence, in articular, is not of finite index. A classical statement in formal language theory (see []) then imlies Corollary 7: If then L is not a regular language. Along the same line as Lemma 6, one can give a more recise statement on suffixes that are owers of a given word. Lemma 8: Let w be in L and w = uv be a roer factorization of w. Then uv k belongs to L only if (k ) v divides π(w) π(u). Proof. The word uv k belongs to L only if ( ) v π(uv k ) π(uv k ) = (π(uv k ) π(uv k )) = ( ) (k ) v = (π(uv) π(u)) is in Z. And this is ossible only if (k ) v divides π(uv) π(u). Lemma 8 will be used in the seuel to show that the closure of L does not contain eventually eriodic infinite words; combined with the classical uming lemma (see []), it imlies another statement related to formal language theory: Corollary 9: If, then L is not a context-free language Suffixes. We observed that L is not suffix-closed. In fact, every word of A is a suffix of some words in L. More recisely, we have the following statement. Proosition : For every integer k and every word w in A k, there exists a uniue integer n, n < k such that w is the suffix of length k of the -exansion of all integers m congruent to n modulo k.

14 66 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. Proof. Given any integer n = n, the reetition k times of the division 4 () yields (4) k n = k n k + k π(a k a k a ). If we do the same for another integer m = m and subtract the euation we get from (4), k (n m ) = k (n k m k ) + k (π(a k a k a ) π(b k b k b )). As k is rime with k, and using Lemma 5, we get (5) n m (mod k ) a k a k a = b k b k b. Since there are exactly k words in A k, each of them must aear once and only once when n ranges from to k and (5) gives the second art of the statement The odometer. Proosition can be interreted, or reformulated, with the construction of a machine that could be called a boosted Pascal machine and would be described as follows. The main feature of the Pascaline, the famous adding machine invented by young Blaise Pascal is a series of toothed wheels linked together by a secial mechanism: when a wheel finishes a full rotation, it sends to the next wheel on the left an imulse that makes the latter move by one unit. Think of the Pascal machine as a series virtually extending infinitely to the left of wheels with a dial in front of each wheel and every dial is marked with digits. The original Pascaline used digits, from to 9, since young Pascal was counting in base, but any will be as good. Let us take = 3 as in our running examle for the remaining of this descrition; the digits are, and. There is an arm, attached to the wheel and moving in front of the corresonding dial. And let us consider the machine as Pascal designed it, but somewhat turned into a clock. In the beginning, every arm is vertical and oints to. Imagine that at every second the rightmost wheel moves by one unit: the arm asses in front of,, again,,, etc. When that arm comes to again, the arm of the second wheel goes to, and so on. After n seconds, one reads on the machine a 4 If n is large enough, this amounts to the first k stes of the MD algorithm. Otherwise, n i = and a i = for i greater than a certain j and this is not, strictly seaking, the MD algorithm.

15 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 67 certain word written on the alhabet A = {,, } which is exactly the writing of n in base 3 if we forget all the s on the left and conversely every word w of A aears exactly once, at time π(w), where π(w) is the number whose writing in base 3 is w. Imagine now that the machine is so to seak boosted and that every uantum of rotation of the wheels is units instead of : the rightmost wheel goes every second from to, from to, from to, etc. and every time the arm of one wheel asses in front of of its dial whether it stos there or not it sends an imulse to the next wheel to the left that makes it moving by units. The seuence of words that will then turn u on the dials of the machine is exactly the -exansions of integers that is the words of L, ordered by length, and within the same length by the lexicograhic order. If a finite such machine were constructed (which is more realistic than an infinite one) with, say, k wheels, the above result states that its behavior is eriodic, of eriod k, and that every ossible configuration of the k wheels will aear once (and only once) during the cycle. The transformation of words witnessed on the boosted Pascal machine at every imulse is the one that is realized by the machine that is usually called the odometer of the number system: it takes as inut a word v reresenting a number n and oututs the word w reresenting the number n +. From the descrition of the boosted Pascal machine, it is easy to build a digit-to-digit right seuential transducer that realizes the odometer for the number system. It is reresented at Figure 3 for our running examle. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (ε, ) (, ) (ε, ) Figure 3. The odometer for the 3 number system.

16 68 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math Order on numbers, order on words. In an integer base, the order on integers and the radix order on the words (on the canonical alhabet) that reresent the numbers coincide, and this is also true of the lexicograhic order on words of the same length (and with a ossible refix in ). The same roerties hold for the number system, rovided only the words in L are considered. Proosition : Let v and w be in L. Then v w if, and only if, π(v) π(w). Proof. Let v = a k a and w = b l b be the -exansions of the integers m = π(v) and n = π(w), resectively. By Theorem, we already know that v = w if, and only if, π(v) = π(w). The roof goes by induction on l, which is (by hyothesis) greater than or eual to k. The roosition holds for l =. Let us write v = a k a and w = b l b, and m = π(v ) and n = π(w ) are integers. It holds: n m = (n m ) + (b a ) Now v w imlies that either v w or v = w and a < b. If v w, then n m by induction hyothesis and thus n m > since b a ( ). If v = w, then n m = (b a ) >. Corollary : Let v and w be in L and only if, π(v) π(w). and of eual length. Then v w if, It is to be noted also that these statements do not hold without the hyothesis that v and w belong to L (to L resectively). For instance, π() = 3/4 < π() = and π() = 7/8 < π() = Conversion between alhabets. Another roerty of the integer base systems that carries over to the number system is the fact that the conversion of digits can be realized by a finite (right) transducer. Let D be a finite alhabet of (ositive or negative) digits that contains A. The digit-set conversion is a ma χ D : D A which commutes to the evaluation ma π, that is, a ma which reserves the numerical value: w D π(χ D (w)) = π(w).

17 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 69 Proosition 3: For any alhabet D the conversion χ D is realizable by a finite letter-to-letter seuential right transducer C D. Proof. Let U D = Z, D A, E, {}, ω be the (infinite) transducer whose set of transitions E is defined by: (6) ( z, (d, a), z ) E z + d = z + a. As z and a are uniuely determined for a given z and d, U D is seuential. If the final function is defined as ω(z) = z for every z in N (and ω(z) =, that is, z is not final, if z < ), it is immediate to verify, by induction on the length of the inut words, that U D, seen as a right transducer, realizes the digit conversion of any word w whose numerical value is ositive. Thus, it remains to show that the accessible art C D of U D is finite. Without loss of generality, one can suose that D is an interval: what matters is the largest digit e and the smallest digit f in D, e and f, at least one of the two ineualities being strict. It follows from (6) that from a state z it is ossible to reach the state z + (res., the state z ) in U D if there exist d in D and a in A such that z = ( (d a) ) /( ) (res., such that z = ( (d a) + ) /( )). Thus the largest accessible ositive state and the smallest accessible state in U D are: z max = e + and z min = and hence C D is finite. f ( ) + = f + The integer addition may be seen after digit-wise addition as a articular case of a digit-set conversion χ D with D = {,,...,( )} and Figure 4 (a) shows the converter that realizes addition in the 3 number system. For reasons which will be exlained in Section 5., we also give at Figure 4 (b) the converter on the alhabet {,,, } in the 3 number system (the signed digit d is denoted d). Remark 4: Let us stress that χ D is defined on the whole set D even for word v such that π(v) is not an integer, and also that, if π(v) is in N, then χ D (v) is the uniue -reresentation of π(v).

18 7 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. (, ) (3, ) (4, ) (, ) (, ) (, ) (, ) (3, ) (4, ) (, ) (3, ) (4, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (a) The converter for the addition. (b) The converter on {,,, }. Figure 4. Two converters for the 3 number system. Remark 5: As is not a Pisot number (when ), the conversion from any reresentation onto the reresentation comuted by the greedy algorithm is not realized by a finite transducer (see [4, Ch. 7]). 4. The tree T The free monoid A is classically reresented as the nodes of the (infinite) full -ary tree: every node is labeled by a word in A and has children, every edge between a node and each of its children is labeled by one of the letter of A and the label of a node is recisely the label of the (uniue) ath that goes from the root to that node. As the language L is refix-closed, it can naturally be seen as a subtree of the full -ary tree, obtained by cutting some edges. This will form the tree T (after we have changed the label of nodes from words to the numbers reresented by these words). This tree, or, more recisely, its infinite aths, will be the basis for the reresentation of reals in the number system. We give now an internal descrition of T, based on the definition of a family of mas from N to N, which will roved to be effective for the study of infinite aths. 4.. Construction of the tree T. Definition 6: (i) For each a in A, let τ a : N N be the artial ma defined by:

19 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 7 n N τ a (n) = (n + a) if (n + a) N undefined otherwise We write d(n) = {a A : τ a (n) is defined}, Md(n) = max{d(n)} for the largest digit for which τ a (n) is defined, and md(n) = min{d(n)} for the smallest digit with the same roerty. (ii) The tree T is the labeled infinite tree (where both nodes and edges are labeled) constructed as follows. The nodes are labeled in N, and the edges in A, the root is labeled by. The children of a node labeled by n are nodes labeled by τ a (n) for a in d(n), and the edge from n to τ a (n) is labeled by a. (iii) We call ath label of a node s of T, and write (s), the label of the ath from the root of T to s. We denote by I the subtree of T made of nodes whose ath label does not begin with a. For examle, the first six levels of T 3 and are shown at Figure 5. I3 The very way T is defined imlies that if two nodes have the same label, they are the root of two isomorhic subtrees of T and it follows from Lemma 6 that the converse is true, that is, two nodes which hold distinct labels are the root of two distinct subtrees of T. As no two nodes of I have the same label, we have Proosition 7: If no two subtrees of I are isomorhic. Definition 6 and the MD algorithm imly directly the following facts that will be used in the seuel, most often without exlicit reference. Lemma 8: For every n in N, it holds: (i) md(n) = d(n) {,,..., } and Md(n) = d(n) {,..., }. (ii) a d(n) and a + A = a + d(n). (iii) a, a + d(n) = τ a+ (n) = τ a (n) +. (iv) md(n + ) = Md(n) + and τ md(n+) (n + ) = τ Md(n) (n) +. And finally: (v) The label of every node s of T is π((s)). In articular, it follows: Corollary 9: n N d = md(n) τ d (n) = n.

20 7 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math Figure 5. The tree T 3, the tree I3 in grey and double edge. This statement induces the definition of the following seuence. Definition : Let (G k ) k N be the seuence of integers defined by: G = and G k+ = G k, k N. It then comes, by induction on k:

21 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 73 Proosition : The nodes of deth k in T, ordered by their ath label in the lexicograhic (or radix) order, are labeled by integers from to G k. We shall return to the comutation of the G k s at Section Minimal and maximal words. The infinite aths in the tree T will be used in Section 5 to define the reresentations of real numbers. Here we consider only some articular infinite aths (or words) in T. We denote by W(n) (res., by w(n)) the label of the infinite ath that starts from a node with label n and that follows always the edges with the maximal (res., minimal) digit label. Such a word is said to be a maximal word (res., a minimal word) in T. The following is a direct conseuence of Lemma 8 (i). Proosition : (i) For all n N, W(n) {,..., } N and w(n) {,..., } N. (ii) Conversely let u be the label of an infinite ath in T. If u is in {,..., } N, then there exists an n such that u = W(n) and if u is in {,..., } N, then there exists an n such that u = w(n). (iii) For every n, the digit-wise difference between w(n + ) and W(n) is ( ) ω. Two secial cases are worth secial notations; we note: t = W() and g = w(). The infinite word t is the maximal element with resect to the lexicograhic order of the label of all infinite aths of T that start from the root. Since τ () =, the infinite word g is the minimal element with resect to the lexicograhic order of the label of all infinite aths of I that start from the root. Notice that, for any rational, ω is the minimal element with resect to the lexicograhic order of the label of all infinite aths of T and that, if =, that is, in an integer base, W(n) = ( ) ω, and w(n) = ω for every n in N. Examle : For = 3, t 3 = and g 3 =. which illustrates, in articular, Proosition (iii).

22 74 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. In Section 3, words of L, and thus labels of finite aths of T, were indexed from right to left or, if one refers, from left to right by decreasing nonnegative integers, always ending with ; the ossibility of extending the indexation to the right after the radix oint, and of using decreasing negative integers, u to minus infinity, for indexing the fractional art of a writing was mentioned at Section.3. When we deal with infinite words that will corresond to reresentations of numbers with only fractional art as it will be the case in the next section we find it much more convenient to change the convention of indexing and use the ositive indices in the increasing order (and starting from ) from left to right. In articular, we write g = g g g 3, and by induction and using Corollary 9 and the fact that the label of a node is the value of its ath label it then comes: Corollary 3: G = π() = and G k = π(g g g k ) for all k N Evaluation of infinite words in T. According to the convention we have just taken on the indexing of infinite words, the evaluation ma takes the following form: (7) a = a a A N π(. a) = ( i a i. ) We use the radix oint. on the left of the infinite word in order to mark the osition of the index and distinguish clearly between the use of the evaluation ma π in euations such as Corollary 3 and (7). Let a = a a be in A N and x = π(. a). With these notations we clearly have: ( ) h (8) x = π(a a a h.a h+ a h+ ), for all h N (9) x = lim h As in an integer base system, we have: i ( ) h π(a a a h ) = lim h Proosition 4: [7] The ma π: A N R is continuous. ( ) h π(a [h]) Notation 5: Let us denote by W the subset of AN that consists of the labels of infinite aths starting from the root of T.

23 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 75 Note that the finite refixes of the elements of W direct conseuence of Lemma 8 is the following. are the words in L. A Proosition 6: If >, then no element of W, but ω, is eventually eriodic. From (8), it then follows: Lemma 7: Let a = a a be in W and x = π(.a). Then, for every k N, ( ) k x =π(a a a k )+ ρ k (x) with ρ k (x)= π(. a k+ a k+ ) <. Proof. The statement holds because π(a a a k ) is an integer as a is in W and the ineuality is strict as no word of W may end in ( )ω. We call branching a node v of T d(π((v))) has at least two elements. if it has at least two children, that is, if Lemma 8: Let v be any branching node in T, and n = π((v)) its label. Let a and b = a + be in d(n) and let m = τ a (n) and m = τ b (n) = m +. Write W(m ) = a a 3 and w(m ) = b b 3. Then, it holds () π(. a a a 3 ) = π(. b b b 3 ). Proof. Proosition (iii) directly yields the comutation: π(. a a a 3 ) π(. b b b 3 ) = and the right member is clearly eual to. ( ( ) + ( ) i We then define the two real numbers ω and γ by: () ω = π(. t ) and γ = π(. g ), ( ) i ) and Lemma 8 imlies: γ = π(. t ) = ω. The next roerty is a kind of a converse of Lemma 8 but a bit more technical. Lemma 9: Suose and let k and r be two integers, k > and r =. Let n be any non negative integer and u and v two words of the

24 76 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. same length l such that π(u) = n and π(v) = n + k. Then ( ) l+r π(.v w(n + k)) π(.u w(n)) ω. Proof. The following notation, insired by Corollary 9, will be convenient: µ(n) = τ md(n) (n) = n for n in N, and of course µ i+ (n) = µ(µ i (n)). The roof goes in three stes. Since n n, the choice of k imlies, for every n in N, (n + k) Thus, since the left handside is an integer, n ( (n + k) n + ) > k. () n N µ(n+k) µ(n)+k+ and i N µ i (n+k) µ i (n)+k+i. It then holds, for every n, m, and z in N: (n + m + z) n + The choice of r imlies then m + z. (k + r) k + + r ( )r ( ) = k + r + k + r. And it then follows, by induction on j, (3) µ j (n + k + r) µ j (n) + G j + k + r, an ineuality that follows from () for j =. Indeed, it holds: µ ( µ j (n) + G j + k + r ) = µj (n) + G j + (k + r) µ j+ (n) + G j + (k + r) µ j+ (n) + G j + k + r. For sake of brevity let us write now a = uw(n) and b = vw(n+k). By definition of w(n) and of µ i (n), it comes π(a [h].) = µ i (n) for h = l+i. Euation (3) may then be rewritten as j N π(b [h].) π(a [h].) G j + k + r with h = l + r + (j ),

25 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 77 and from (9) it follows π(.b) π(.a) lim j + = ( ( ) l+r+j (G j + k + r) ) [ l+r ( ) j lim G j + j ]. Figure 6 shows the tree T 3 again. But this time every node s is given an ordinate eual to π((s)); nodes at the same level in the tree are given the same abscissa (as in Figure 5) and the distance between two levels of the tree is multilied by when the level increases, which gives the fractal asect The Josehus Problem. The above definitions and notations allow us to give another exression for the integer seuence (G k ) k N. Proosition 3: For every k in N, there exists an integer e k, e k <, such that G k = Proof. From the definition of γ γ γ ( ) k+ e k. and as in Lemma 7, it follows: ( ) k+ = π( g g k.) + π(.g k+ g k+ ) = G k + e k, where e k is an integer strictly smaller than since g i is in {,..., } for every i. Corollary 3: If then, for every k in N, G k = ( γ ) k+. Remark 3: Still in the case where, then for every k, the digit g k of the -exansion of γ is obtained as follows: (i) comute G k+ = G k (ii) g k+ = G k+ (mod ). The definition of the seuence G k and the comutation of γ have been develoed not only because they are imortant for the descrition of T but also as they relate to a classical roblem in combinatorics. Insired by the so-called Josehus roblem, Odlyzko and Wilf consider, for a real α >, the iterates of the function f(x) = α x : f = and f n+ = α f n for n. They show (in [7]) that in the cases where α, or α = / for

26 78 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math ω 3 Figure 6. Another view on T 3 (with four more levels) some integer, then there exists a constant H(α) such that f n = H(α)α n for all n.

27 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 79 Thus, we obtain the same result as in [7] for any rational α =, with, and we find H( ) = γ = ω. Our method does not yield an indeendent way of comuting this constant, as was called for in [7], but the -exansion of ω gives at least an easy algorithm. In the case where = (the Josehus case), the constant ω is the constant K() in [7]. In this case the integer e k of Proosition 3 is less than, and this is the same bound as in [7]. Examle 3: For = 3, the constant ω 3 is the constant K(3) already discussed in [7,, 4]. Its decimal exansion: ω 3 = is recorded as Seuence A8386 in [3]. Observe that, in the same case, the seuence (G k ) k is Seuence A649 in [3]. 5. Reresentation of the reals Every infinite word a in A N is given a real value x by π: x = π(.a), and a is called a -reresentation of x. Our urose here is to associate with every real number a -reresentation which will be as canonical as ossible. In contrast with what is done in Pisot base number systems, where the canonical reresentation, the greedy exansion, is defined by an algorithm which comutes it for every real, we set these canonical -exansions a riori. Then we have to rove: first, that they reresent indeed the reals and, second, to what extent they are canonical. In a second art, we give an algorithmic way to comute a -reresentation which we call the comanion -reresentation. This reresentation is not on the alhabet A anymore but on a larger alhabet with negative digits. We investigate then how one can recover the -exansion from the comanion - reresentation and it is from their relationshis that we shall derive in the next section the new results on the ower of rational numbers. 5.. The -exansions of real numbers. As announced, the set of - exansions is defined a riori and not algorithmically. Definition 33: The set of exansions in the number systems is W.

28 8 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. In other words, an element a of W is a -exansion of the real x = π(.a) and conversely any element of A N which does not belong to W is not a -exansion. The following Lemma 34 and Theorem tell resectively that -exansions are not too many nor too few and vindicate the definition. Lemma 34: The ma π: W R is order reserving. Proof. Let a and b be in W. If a b then, for every k in N, a a a k b b b k and then, by Corollary, π(a a a k ) π(b b b k ). By (9), π(. a) π(. b). By contrast, it follows from the examles given after Corollary that the ma π: A N R is not order reserving. Let X = π(w ). The elements of X are nonnegative real numbers less than or eual to ω : X [, ω ] (note that ω < ). Theorem : Every real in [, ω ] has at least one -exansion, that is, X = [, ω ]. Proof. By definition, the set W is the set of infinite words w in AN such that any refix of w is in L. As L is refix-closed since L is refixclosed and the emty word belongs to L W is closed (see [9]) in the comact set A N, hence comact. Since π is continuous, X is closed. Suose that [, ω ] \ X is a nonemty oen set, containing a real u. Let y = su{x X : x < u} and z = inf{x X : x > u}. Since X is closed, y and z both belong to X. Let a = a a be the largest -exansion of y and b = b b the smallest -exansion of z (in the lexicograhic order). Of course, a b since a b. Let a a N be the longest common refix of a and b (with the convention that N can be ). Set m = π(a a N.), n = π(a a N a N+.) and = π(a a N b N+.). Then a a a N a N+ W(n) a a N b N+ w() b. By the choice of b, π(.a a N a N+ W(n)) < z, and by the choice of a, a = a a N a N+ W(n). Symmetrically, b = a a N b N+ w(). If a N+ + < b N+, then there exists a digit c in d(m) such that a N+ + c < b N+. For any c in A N such that c = a a N c c is in W (and there exist some), we have a c b.

29 Vol. 68, 8 RATIONAL BASE NUMBER SYSTEMS 8 Whatever the value of π(.c), y or z, we have a contradiction with the extremal choice of a and b. If a N+ + = b N+, then = n + and z = y by Lemma 8, hence a contradiction. And thus X = [, ω ]. A word in W is said to be eventually maximal (res., eventually minimal) if it has a suffix which is a maximal (res., minimal) word. From Lemma 8, follows Proosition 35: If a in W is eventually maximal, then x = π(.a) has another -exansion which is eventually minimal, and conversely. Theorem 36: The set of reals in X that have more than one -exansion is countably infinite. The -exansions of such reals are eventually maximal or eventually minimal. We have seen, with Lemma 8, that to every branching node in T corresonds a real with at least two -exansions. The theorem will thus be established when the following roosition will be roved. Proosition 37: Let x be a real in [, ω ], with more than one -exansion. Then x has at most k + -exansions, where k is the least integer strictly greater than, and can be associated with at most k branching nodes of T. The smallest of these exansions is eventually maximal and the largest eventually minimal; the others, if any, are both eventually maximal and eventually minimal. Proof. Let R = π (x) W be the (closed) set of -exansions of x. Let a = a a be the smallest and b = b b the largest -exansion of x (in the lexicograhic order) and, as above, let a a N be the longest common refix of a and b (with the convention that N can be ). Let c = c c be in R and different from (and thus smaller than) b. We first claim that it does not exist any integer h such that π(b [h].) π(c [h].) k +. Suose the contrary, write π(c [h].) = n, π(b [h].) = m n + k and let d be the word of L of length h such that π(d) = n. It then holds c d w(n) b [h] w(m) b,

30 8 S. AKIYAMA, CH. FROUGNY AND J. SAKAROVITCH Isr. J. Math. and thus, by Lemma 9, π(.c) π(.d w(n)) < π(.b [h] w(m)) π(.b), which contradicts π(.c) = π(.b) = x. This directly imlies that R, which consists of the c in W such that a c b, contains at most k + elements and the corresonding subtree of T at most k branching nodes. Suose now that for an integer h greater than N, c h+ is not the maximal digit, that is, c h+ is smaller than Md(π ( c [h]. ) ). Let us write c h = W(π ( ) c [h] ). It then holds: c c [h] c h b. From this we deduce that the seuence of integers h such that c h+ is not the maximal digit is finite (and smaller than k) and thus that c is eventually maximal. Symmetrically, any -exansion in R that is different from (and thus larger than) a is eventually minimal. It follows in articular that if then no real number has more than two -exansions. A simle combinatorial argument allows to widen the condition and to recover the case = 3. Corollary 38: If, then no real number has more than two - exansions. Proof. Suose = since the other cases are already settled by Proosition 37. If x has more than two -exansions, then by Proosition 37 one is both eventually maximal and eventually minimal and thus eventually written (cf. Proosition ) on the alhabet: {,..., } {,..., } = { } reduced to one letter, since =. Contradiction, since no -exansion is eventually eriodic. Remark 39: In contrast with the classical reresentations of reals, the finite refixes of a -exansion of a real number, comleted by zeroes, are not - exansions of real numbers (though they can be given a value by the function π of course), that is to say, if a non emty word w is in L, then the word w ω does not belong to W.