Minimal weight expansions in Pisot bases

Minimal weight expansions in Pisot bases Christiane Frougny and W olfgang Steiner Abstract. For applications to cryptography , it is imp ortant to represent numbers with a small number of non-zero digits (Hamming weight) or wi th small absolute sum of digits. The problem of fi nding representations with minimal weight has been solved for integ er bases, e.g. by the non-adjace nt form in base 2. In this pa per , we consider numeration systems with respe ct to real bases β which are Pisot numbers and prov e that the expansions with minimal absolute sum of digits are recognizable by finite automata. When β is t he Golden R atio, t he Tribon acci number or the smallest Pisot number , we determine e xpansion s with minimal nu mber of digits ± 1 and gi ve e xplicitely the finite au tomata recognizing all these expa nsions. The av erage weight is lo wer than for the non-adjacen t form. K eywords. Minimal weight, beta-expan sions, P isot numbers, Fibonacci numbers, automata. AMS classification. 11A63, 11B39, 68Q45, 94A60. 1 Intr oduction Let A be a set of (integer) digits and x = x 1 x 2 · · · x n be a w ord with letters x j in A . The weight of x is the absolute sum of digits k x k = P n j = 1 | x j | . The Hamming w eight of x is the number of non-zero digits in x . Of course, when A ⊆ {− 1 , 0 , 1 } , the two definitions coincide. Expansions of minimal we ight in intege r bases β ha ve be en studied e xtensi vely . When β = 2, it is known since Booth [ 4] and Reitwiesner [23] how to obtain such an expansion with the digit set { − 1 , 0 , 1 } . The well-known non-adjacent form (NAF) is a particular expansion of minimal weight with the property that the non-zero digits are isolated. I t has many applications to cryptography , s ee in particular [20, 17, 21]. Other expansions of minimal weight in integer base are studied in [14, 16]. Er godic properties of signed binary ex pansions are established in [6]. Non-standard nu mber systems — where the base is not an integer — ha ve been studied from various points of view . Expansions in a r eal non-integral base β > 1 ha ve been introduced by Rényi [24] and s tudied initially by Parry [22]. Number the- oretic tr ansforms where numbers are represented in base the Golden Ratio ha ve been introduced in [7] for applic ation to signal pr ocessing and fast con v o lution. Fibonacci representations hav e been used in [19] to design expone ntiation algorithms based on addition chains. Recen tly , the inv estigation of minimal weight expansions has been extend ed to the Fibonacci numeration system by Heuber ger [ 15], who ga ve an equi v- alent to the N AF . Solinas [26] has shown how to represent a scalar in a complex base τ related to Kob litz curv es, and has give n a τ -N AF form, and the Hamming weight of these representations has been studied in [9]. This researc h wa s supported by the French Agence National e de la Reche rche, grant ANR-JCJC06-134288 “DyCoNum”. 2 Christiane Frougny and W olfgang Steiner In this paper , we study expansions in a real base β > 1 which is not an integer . Any number z in the interv al [ 0 , 1 ) has a so-called greed y β -e xpansion giv en by the β - transformation τ β , which relies on a greedy algorithm: let τ β ( z ) = β z − ⌊ β z ⌋ and define, for j ≥ 1, x j = ⌊ β τ j − 1 β ( z ) ⌋ . Then z = P ∞ j = 1 x j β − j , where the x j ’ s are integer digits in the alphabet { 0 , 1 , . . . , ⌊ β ⌋ } . W e write z = . x 1 x 2 · · · . If there exists a n such that x j = 0 for all j > n , the expansion is s aid to be finite and we write z = . x 1 x 2 · · · x n . By shifting, any non-negati ve real numbe r has a greedy β -expansion: If z ∈ [ 0 , β k ) , k ≥ 0, and β − k z = . x 1 x 2 · · · , then z = x 1 · · · x k . x k + 1 x k + 2 · · · . W e consider the sequences of digits x 1 x 2 · · · as words. Since we want to minimize the weight, we are only interested in finite words x = x 1 x 2 · · · x n , but we allo w a priori arbitrary digits x j in Z . The corresponding set of numbers z = . x 1 x 2 · · · x n is therefore Z [ β − 1 ] . Note that we do not require that the greedy β -expan sion of every z ∈ Z [ β − 1 ] ∩ [ 0 , 1 ) is finite, although this property ( F) holds for the three numbers β studied in Sections 4 to 6, see [12, 1]. The set of finite words with letters in an alphabet A is denoted by A ∗ , as usual. W e define a relation on w ords x = x 1 x 2 · · · x n ∈ Z ∗ , y = y 1 y 2 · · · y m ∈ Z ∗ by x ∼ β y if and only if . x 1 x 2 · · · x n = β k × . y 1 y 2 · · · y m for some k ∈ Z . A word x ∈ Z ∗ is said to be β -heavy if there exists y ∈ Z ∗ such that x ∼ β y and k y k < k x k . W e say that y is β -lighter than x . This means that an appropriate shift of y provides a β -expansion of the number . x 1 x 2 · · · x n with smaller absolute s um of digits than k x k . If x is not β -hea vy , then we call x a β -ex pansion of minimal weight . It is easy to see that e very word containing a β -hea vy fac tor is β -hea vy . Therefore we can restrict our attention to strictly β -heavy words x = x 1 · · · x n ∈ Z ∗ , which means that x is β -heavy , and x 1 · · · x n − 1 and x 2 · · · x n are not β - hea vy . In the followin g, we consider real bases β satisfying the condition ( D B ) : there exists a w ord b ∈ { 1 − B , . . . , B − 1 } ∗ such that B ∼ β b and k b k ≤ B for s ome positiv e integer B . Corollary 3.2 and Remark 3.4 show that eve ry class of words (with respect to ∼ β ) contains a β -expa nsion of minimal weight with digits in { 1 − B , . . . , B − 1 } if and only if β satisfies (D B ). If β is a Pisot number , i.e., an algebraic integer greater than 1 such that all the other roots of its minimal polynomial are in modulus less than one, then it satisfies (D B ) for some B > 0 by Proposition 3.5. The contrary is n ot true: There exist algebraic integers β > 1 sats fying (D B ) which are not Pisot, e.g. the positive solution of β 4 = 2 β + 1 is not a Pisot number b ut satisfies (D 2 ) since 2 = 1000 . ( − 1 ) . The follo wing example provides a la rge class of numbers β s atisfying (D B ). Example 1.1. I f 1 = . t 1 t 2 · · · t d ( t d + 1 ) ω with integers t 1 ≥ t 2 ≥ · · · ≥ t d > t d + 1 ≥ 0, then β satisfies (D B ) with B = t 1 + 1 = ⌊ β ⌋ + 1, since β d + 1 − t 1 β d − · · · − t d β − t d + 1 = t d + 1 β − 1 = β d − t 1 β d − 1 − · · · − t d and thus β d + 1 − ( 1 + t 1 ) β d + ( t 1 − t 2 ) β d − 1 + · · · + ( t d − 1 − t d ) β + ( t d − t d + 1 ) = 0 . Minimal weight expa nsions in Pisot bases 3 Recall that the set of greedy β -expansions is recognizable by a finite automaton when β is a Pisot number [3]. In this w ork, we prove that the set of all β -expansions of minimal weight is recognized by a finite automaton when β is a Pisot numbe r . W e then consider particular Pisot numbers satisfying (D 2 ) which ha ve been exten - siv ely studied from various points of view . When β is the Golden Ratio, we construct a finite transducer which give s , for a strictly β -heavy word as input, a β -lighter word as outpu t. Similarly to th e Non-Adja cent Form in ba s e 2, w e define a particular u nique expan sion of minimal weight av oiding a certain giv en set of factors. W e show that there is a finite tr ansducer which conv erts all w ords of minimal weight into these ex- pansions a voiding these factors. From these transducers, we derive the minimal au- tomaton r ecognizing the set of β -expansions of minimal weight in {− 1 , 0 , 1 } ∗ . W e gi ve a branching transformation which pro vides all β -expansions of minimal weig ht in {− 1 , 0 , 1 } ∗ of a giv en z ∈ Z [ β − 1 ] . Similar results are obtained for the representation of integers in the Fibonacci numera tion system. The a verage weight of ex pansions of the numbers − M , . . . , M is 1 5 log β M , w hich mea ns that typic ally only every fifth digit is non-zero. Note that the corresponding v alue f or 2-expansions of minimal weigh t is 1 3 log 2 M , see [2, 5], and that 1 5 log β M ≈ 0 . 288 log 2 M . W e obtain s imilar results for the case where β is the so-called T ribonacci number , which s atisfies β 3 = β 2 + β + 1 ( β ≈ 1 . 839), and the corresponding representa- tions for inte gers. In this case, the averag e weight is β 3 β 5 + 1 log β M ≈ 0 . 282 log β M ≈ 0 . 321 log 2 M . Finally we consider the smallest Pisot num ber , β 3 = β + 1 ( β ≈ 1 . 325), which pro- vides representations of inte gers with ev en lo wer weight than the Fibonacci numeration system: 1 7 + 2 β 2 log β M ≈ 0 . 095 log β M ≈ 0 . 234 log 2 M . Since the proof techniques for the T ribonacci number and the smallest Pis ot number are quite similar to the Golden Ratio case (but more complicated), s ome parts of the proofs are not contained in the final version of this paper . The interested reader can find them in [13]. 2 Pr eliminaries A finite sequence of elements of a set A is called a word , and the set of words on A is the fr ee monoid A ∗ . The s et A is called alphabet . The set of infinite sequences or infinite words on A is denoted by A N . Let v be a word of A ∗ , denote by v n the concatena tion of v to itself n times, and by v ω the infinite concaten ation v vv · · · . A finite w ord v is a factor of a (finite or infinite) word x if there exists u and w s uch that x = uv w . When u is the empty word, v is a pr efix of x . The prefix v is strict if v 6 = x . When w is empty , v is said to be a suffix of x . W e recall some definitions on automata, see [10] and [25] for instance. An automa - ton o ver A , A = ( Q, A, E , I , T ) , is a d irected graph labelled by ele ments of A . The set of vertices, traditionally calle d states , is denoted by Q , I ⊂ Q is the set of initial states, T ⊂ Q is the set of terminal states and E ⊂ Q × A × Q is the set of labelle d edges . If ( p, a, q ) ∈ E , we write p a → q . The automa ton is finite if Q is finite. A subset H of A ∗ is said to be r ecogn izable by a finite automaton if there exists a finite automaton A 4 Christiane Frougny and W olfgang Steiner such that H is equal to the set of labels of paths starting in an initial state and ending in a terminal state. A transducer is an automaton T = ( Q , A ∗ × A ′∗ , E , I , T ) where the edges of E are labelled by couples of words in A ∗ × A ′∗ . It is said to be finite if the set Q of states and the set E of edges are finite. If ( p, ( u, v ) , q ) ∈ E , we write p u | v − → q . In this pape r we consider letter-to-letter transducers, where the edges are labelled by elements of A × A ′ . The input automaton of such a transducer is obtained by taking the projectio n of edges on the first compone nt. 3 General case In this section, our aim is to prove that one can construct a finite au tomaton reco gnizing the set of β -expansions of minimal weight when β is a Pisot number . W e need first some combina torial results for bases β satisfying (D B ). Note that β is not assumed to be a Pisot number here. Pr oposition 3.1. If β satisfies (D B ) with some inte ger B ≥ 2 , then for every word x ∈ Z ∗ ther e ex is ts some y ∈ { 1 − B , . . . , B − 1 } ∗ with x ∼ β y and k y k ≤ k x k . Cor ollary 3.2. If β satis fies (D B ) with some integ er B ≥ 2 , th en for e very wor d x ∈ Z ∗ ther e ex is ts a β -e xpansion of minimal weight y ∈ { 1 − B , . . . , B − 1 } ∗ with x ∼ β y . Remark 3.3. If β satisfies (D B ) for some positiv e integer B , then it is easy to see that β satisfies (D C ) for e very integer C > B . Remark 3.4. If β does not satisfy (D B ), then all w ords x ∈ { 1 − B , . . . , B − 1 } ∗ with x ∼ β B are β -hea vier than B . It follows that the set of β -expansions of minimal weight x ∼ β B is 0 ∗ B 0 ∗ . Pr oof of Pr oposition 3.1. Let A = { 1 − B , . . . , B − 1 } . If x = x 1 x 2 · · · x n ∈ A ∗ , then there is nothing to do. Otherwise, we use (D B ): there exists some word b = b − k · · · b d ∈ A ∗ such that b − k · · · b − 1 ( b 0 − B ) b 1 · · · b d ∼ β 0 and k b k ≤ B . W e use this relation to decrease the absolute v alue of a digit x h 6∈ A without increasing the weight of x , and we s ho w that we eve ntually obtain a word in A ∗ if we always choose the rightmost such digit. M ore precisely , set x ( 0 ) j = x j for 1 ≤ j ≤ n , x ( 0 ) j = 0 for j ≤ 0 and j > n , b j = 0 for j < − k and j > d . Define, recursive ly for i ≥ 0, h i = max { j ∈ Z : | x ( i ) j | ≥ B } , x ( i + 1 ) h i = x ( i ) h i + sgn ( x ( i ) h i )( b 0 − B ) , x ( i + 1 ) h i + j = x ( i ) h i + j + sgn ( x ( i ) h i ) b j for j 6 = 0 , as long as h i exists. Then we ha ve P j ∈ Z | x ( 0 ) j | = k x k , P j ∈ Z x ( i + 1 ) j β − j = P j ∈ Z x ( i ) j β − j and X j ∈ Z | x ( i + 1 ) j | = | x ( i + 1 ) h i | + X j 6 = 0 | x ( i + 1 ) h i + j | ≤ | x ( i ) h i | + | b 0 | − B + X j 6 = 0 ( | x ( i ) h i + j | + | b j | ) ≤ X j ∈ Z | x ( i ) j | . Minimal weight expa nsions in Pisot bases 5 If h i does not exist, then we hav e | x ( i ) j | < B for all j ∈ Z , and the sequence ( x ( i ) j ) j ∈ Z without the leading and trailing zeros is a word y ∈ A ∗ with the desired properties. Since k x k is finite, we hav e P j ∈ Z | x ( i + 1 ) j | < P j ∈ Z | x ( i ) j | only for finitely many i ≥ 0. In particular , the algorithm terminates after at most k x k − B + 1 steps if k b k < B . 1 If k b k = B and P j ∈ Z | x ( i + 1 ) j | = P j ∈ Z | x ( i ) j | , then we ha ve h i − 1 X j = −∞ | x ( i + 1 ) j | = h i − 1 X j = −∞ | x ( i ) j | + k X j = 1 | b − j | and ∞ X j = h i + 1 | x ( i + 1 ) j | = ∞ X j = h i + 1 | x ( i ) j | + d X j = 1 | b j | . Assume that h i exists for all i ≥ 0. If ( h i ) i ≥ 0 has a minimum, then there exists an increasing sequence of indices ( i m ) m ≥ 0 such tha t h i m ≤ h ℓ for all ℓ > i m , m ≥ 0, thus k x k ≥ h i m − 1 X j = −∞ | x ( i m + 1 ) j | ≥ h i m − 1 − 1 X j = −∞ | x ( i m − 1 + 1 ) j | + k X j = 1 | b − j | ≥ · · · ≥ ( m + 1 ) k X j = 1 | b − j | . If P k j = 1 | b − j | > 0, this is not possible since k x k is finite. Similarly , ( h i ) i ≥ 0 has no maximum if P d j = 1 | b j | > 0. Since x ( i + 1 ) j can differ from x ( i ) j only for h i − k ≤ j ≤ h i + d , we have h i + 1 ≤ h i + d for all i ≥ 0. If h i < h i ′ , i < i ′ , then there is therefore a sequence ( i m ) 0 ≤ m ≤ M , i ≤ i 0 < i 1 < · · · < i M = i ′ , with M ≥ ( h i ′ − h i ) /d such that h i m ≤ h ℓ for all ℓ ∈ { i m , i m + 1 , . . . , i ′ } , m ∈ { 0 , . . . , M } . As abov e, we obtain k x k ≥ ( M + 1 ) P k j = 1 | b − j | , b ut M can be arbitrarily lar ge if ( h i ) i ≥ 0 has neither minimum nor maximum. Hence we ha ve s ho wn that h i cannot exist for all i ≥ 0 if P k j = 1 | b − j | > 0 and P d j = 1 | b j | > 0. It remains to consider the case k b k = B with k = 0 or d = 0. Assume, w . l.o.g., d = 0. T hen we hav e h i + 1 ≤ h i . If h i exists for all i ≥ 0, then both P k j = 0 | x ( i ) h i − j | and P ∞ j = 1 | x ( i ) h i + j | are ev entually constant. Therefore we must ha ve s ome i, i ′ with h i ′ < h i such that x ( i ′ ) h i ′ − k · · · x ( i ′ ) h i ′ = x ( i ) h i − k · · · x ( i ) h i , x ( i ′ ) h i ′ + 1 x ( i ′ ) h i ′ + 2 · · · = 0 h i − h i ′ x ( i ) h i + 1 x ( i ) h i + 2 · · · , and x ( i ) h i − j = x ( i ′ ) h i ′ − j = 0 for all j > k . T his implies x ( i ) h i − k · · · x ( i ) h i ∼ β 0 or β h i − h i ′ = 1. In the first case, x ( i ) h i + 1 x ( i ) h i + 2 · · · without the trailing zeros is a word y ∈ A ∗ with the desired properties. In the latter case, each x ∈ Z ∗ can be easily transformed into s ome y ∈ {− 1 , 0 , 1 } ∗ with y ∼ β x and k y k = k x k , and the proposition is prov ed. ✷ The follow ing proposition shows slightly more than the existence of a positi ve inte- ger B s uch that β satisfies (D B ) when β is a Pisot number . Pr oposition 3.5. F or e very Pisot number β , ther e exists some positive inte ger B and some wor d b ∈ Z ∗ such tha t B ∼ β b and k b k < B . Pr oof. If β is an integer , then we can choose B = β and b = 1. So let β be a Pis ot number of degree d ≥ 2, i. e., β has d − 1 Galois conjugates β ( j ) , 2 ≤ j ≤ d , with | β ( j ) | < 1. For e very z ∈ Q ( β ) set z ( j ) = P ( β ( j ) ) if z = P ( β ) , P ∈ Q [ X ] . 1 For the proof of T heorem 3.11, it is sufficie nt to consider the case k b k < B . Howe ver , Corollary 3.2 is partic ularly interesting in the case k b k = B , and we use it in the follo wing sections for B = 2. 6 Christiane Frougny and W olfgang Steiner Let B be a positive integer , L = ⌈ log B / log β ⌉ , and x 1 x 2 · · · the greedy β -exp ansion of z = β − L B ∈ [ 0 , 1 ) . Since τ k β ( z ) = β τ k − 1 ( z ) − x k = · · · = β k z − k X ℓ = 1 x ℓ β k − ℓ , we ha ve    ( τ k β ( z )) ( j )    =     ( β ( j ) ) k z ( j ) − k X ℓ = 1 x ℓ ( β ( j ) ) k − ℓ     <   β ( j )   k   z ( j )   + ⌊ β ⌋ 1 − | β ( j ) | for all k ≥ 0 and 2 ≤ j ≤ d . Set k = max 2 ≤ j ≤ d ⌈− log | z ( j ) | / log | β ( j ) |⌉ . T hen τ k β ( z ) is an element of the finite set Y =  y ∈ Z [ β − 1 ] ∩ [ 0 , 1 ) : | y ( j ) | < 1 + ⌊ β ⌋ 1 − | β ( j ) | for 2 ≤ j ≤ d  . For e very y ∈ Y , we can choose a β -expan sion y = . a 1 · · · a m . Let W be the maximal weight of all these e xpansions and τ k β ( z ) = . a ′ 1 · · · a ′ m . Since z = . x 1 . . . x k + τ k β ( z ) , the digitwise addition of x 1 · · · x k and a ′ 1 · · · a ′ m provides a w ord b with b ∼ β B and k b k ≤ k ⌊ β ⌋ + W = max 2 ≤ j ≤ d  log B log β  − log B log | β ( j ) |  ⌊ β ⌋ + W = O ( log B ) . If B is sufficiently lar ge, we hav e therefore k b k < B . ✷ In order to understand the relation ∼ β on A ∗ , A = { 1 − B , . . . , B − 1 } , we hav e to consider the words z ∈ ( A − A ) ∗ with z ∼ β 0. Therefore we set Z β = n z 1 · · · z n ∈ { 2 ( 1 − B ) , . . . , 2 ( B − 1 ) } ∗    n ≥ 0 , n X j = 1 z j β − j = 0 o and recall a r esult from [11]. All the automata considered in this paper process w ords from left to right, that is to say , most significant digit first. Lemma 3.6 ([11]). If β is a Pisot numbe r , then Z β is re cognized by a finite automaton . For conv enience, we quickly explain the construction of the automaton A β recogniz- ing Z β . The state s of A β are 0 and all s ∈ Z [ β ] ∩ ( 2 ( 1 − B ) β − 1 , 2 ( B − 1 ) β − 1 ) which are a ccessible from 0 by paths consisting of transitions s e → s ′ with e ∈ A − A such that s ′ = β s + e . The state 0 is both initial and terminal. When β is a Pisot number , then the set of states is finite. Note that A β is s ymmetric, meaning that if s e → s ′ is a transition, then − s − e → − s ′ is also a transition. The automaton A β is accessible and co-accessible. The re dundancy automaton (or tr ansducer) R β is similar to A β . Each transition s e → s ′ of A β is replaced in R β by a set of transitions s a | b − → s ′ , with a, b ∈ A and a − b = e . From Lemma 3.6, one obtains the followin g lemma. Minimal weight expa nsions in Pisot bases 7 Lemma 3.7. The r edundancy transducer R β r ecognizes the set  ( x 1 · · · x n , y 1 · · · y n ) ∈ A ∗ × A ∗   n ≥ 0 , . x 1 · · · x n = . y 1 · · · y n  . If β is a Pisot numbe r , then R β is finite. From the redundanc y transducer R β , one constructs another transducer T β with states of the form ( s, δ ) , where s is a state of R β and δ ∈ Z . The transitions are of the form ( s, δ ) a | b − → ( s ′ , δ ′ ) if s a | b − → s ′ is a tr ansition in R β and δ ′ = δ + | b | − | a | . T he initial state is ( 0 , 0 ) , and terminal states are of the form ( 0 , δ ) with δ < 0. Lemma 3.8. The transduc er T β r ecognizes the set  ( x 1 · · · x n , y 1 · · · y n ) ∈ A ∗ × A ∗   . x 1 · · · x n = . y 1 · · · y n , k y 1 · · · y n k < k x 1 · · · x n k  . Of co urse, the tr ansducer T β is not finite, and th e core of the proof of the m ain re sult consists in sho wing that we need only a finite part of T β . W e also need the follo wing well-kno wn lemma, and giv e a proof for it beca us e the construction in the proof will be used in the follo wing sections. Lemma 3. 9. Let H ⊂ A ∗ and M = A ∗ \ A ∗ H A ∗ . If H is r ecognized by a finite automaton, then so is M . Pr oof. Suppose that H is recognized by a finite automaton H . Let P be the set of strict prefixes of H . W e construct the minimal automaton M of M as follo ws. The set of states of M is the quotient P / ≡ where p ≡ q if the paths labelled by p end in the same set of states in H as the paths labelle d by q . Since H is finite, P / ≡ is finite. T ransitions are defined as follows. Let a be in A . If p a is in P , then there is a transition [ p ] ≡ a → [ pa ] ≡ . If pa is not in H ∪ P , then there is a transition [ p ] ≡ a → [ v ] ≡ with v in P maximal in length such that pa = uv . Every state is termin al. ✷ No w , we can prove the f ollo wing theorem. The main result, Theorem 3.11, will be a special case of it. Theor em 3.10. Let β be a Pisot number and B a positive inte ger such that (D B ) holds. Then one can construct a finite automaton r ecognizing the s et of β -ex pansions of minimal weight in { 1 − B , . . . , B − 1 } ∗ . Pr oof. Let A = { 1 − B , . . . , B − 1 } , x ∈ A ∗ be a s trictly β -hea vy word and y ∈ A ∗ be a β - expan sion of minimal weight with x ∼ β y . Such a y exists because of Proposition 3.1. Extend x, y to words x ′ , y ′ by adding leading and trailing zeros such that x ′ = x 1 · · · x n , y ′ = y 1 · · · y n and . x 1 · · · x n = . y 1 · · · y n . Then there is a path in the transducer T β composed of transitions ( s j − 1 , δ j − 1 ) x j | y j − → ( s j , δ j ) , 1 ≤ j ≤ n , with s 0 = 0, δ 0 = 0, s n = 0, δ n < 0. W e determine bounds for δ j , 1 ≤ j ≤ n , which depend only on the state s = s j . Choose a β -e xpansion of s , s = a 1 · · · a i . a i + 1 · · · a m , and set w s = k a 1 · · · a m k . If δ j > w s , then we hav e k y 1 · · · y j k > k x 1 · · · x j k + w s . Since s j = ( x 1 − y 1 ) · · · ( x j − y j ) . , 8 Christiane Frougny and W olfgang Steiner the digitwise subtraction of 0 max ( i − j, 0 ) x 1 · · · x j 0 m − i and 0 max ( j − i, 0 ) a 1 · · · a m provides a word which is β -lighter than y 1 · · · y j , which contradicts the assumption that y is a β - expa nsion of minimal weight. Let W = max { w s | s is a state in A β } . If δ j ≤ − W − B , then let h ≤ j be such that x h 6 = 0, x i = 0 for h < i ≤ j . Since | x h | < B , we hav e δ h − 1 < δ j + B ≤ − W ≤ − w s h − 1 , hence k x 1 · · · x h − 1 k > k y 1 · · · y h − 1 k + w s h − 1 . Let a 1 · · · a m be the word which was used for the definition of w s h − 1 , i.e., s h − 1 = a 1 · · · a i . a i + 1 · · · a m , w s h − 1 = k a 1 · · · a m k . Then the digitwise addition of 0 max ( i − h + 1 , 0 ) y 1 · · · y h − 1 0 m − i and 0 max ( h − 1 − i, 0 ) a 1 · · · a m provides a word which is β - lighter than x 1 · · · x h − 1 . Since x h 6 = 0, this contradicts the assumption that x is strictly β -hea vy . Let S β be the restriction of T β to the states ( s, δ ) with − W − B < δ ≤ w s with some additional initial and terminal states: Every state which can be reached f rom ( 0 , 0 ) by a path with input in 0 ∗ is initial, and e very state with a path to ( 0 , δ ) , δ < 0, with an input in 0 ∗ is terminal. Then the set H which is recognized by the input automaton of S β consists only of β -hea vy words and contains all strictly β -hea vy words in A ∗ . Therefore M = A ∗ \ A ∗ H A ∗ is the s et of β -expansions of minimal weight in A ∗ , and M is recognizable by a finite automaton by Lemma 3.9. ✷ Theor em 3. 11. Let β be a Pisot number . Then one can construct a finite automaton r ecognizing the set of β -expansions of minimal weight. Pr oof. Proposition 3.5 s ho ws that β satisfies (D B ) for some positiv e integer B , and that no β -expansion of minimal weight y ∈ Z ∗ can contain a digit y j with | y j | ≥ B , since we obtain a β -lighter w ord if we rep lace B by b as in the proof of Pr oposition 3. 1. Therefore Theorem 3.10 implies Theorem 3.11. ✷ 4 Golden Ratio case In this section we give e xplicit constructions f or the case where β is the Golden Ratio 1 + √ 5 2 . W e ha ve 1 = . 11, hence 2 = 10 . 01 and β satisfies (D 2 ), see also Example 1.1. Corollary 3.2 s ho ws that e very z ∈ Z [ β − 1 ] can be represented by a β - expan sion of minimal weight in {− 1 , 0 , 1 } ∗ . For most applications, only these expa nsions are in- teresting. Remark that the digits of arbitrary β -expan sions of minimal weight are in {− 2 , − 1 , 0 , 1 , 2 } by the proof of Theorem 3.11, s ince 3 = 100 . 01. For typog raphical reasons, we write the digit − 1 as ¯ 1 in words and transitions. 4.1 β -expansio ns of minimal weight f or β = 1 + √ 5 2 Our aim in this section is to construct explicitly the finite automaton recognizing the β - expa nsions of minimal weight in A ∗ , A = { − 1 , 0 , 1 } . Theor em 4. 1. If β = 1 + √ 5 2 , then the set of β -e xpansions of minimal weight in {− 1 , 0 , 1 } ∗ is r ecognized by the finite automaton M β of F igur e 1 where all states are te r minal. Minimal weight expa nsions in Pisot bases 9 0 1 ¯ 1 0 0 0 0 1 ¯ 1 ¯ 1 1 0 0 1 ¯ 1 0 0 0 10 ¯ 10 010 0 ¯ 100 0 ¯ 10 0100 00 00 100 ¯ 100 Figur e 1. Automaton M β recognizing β -expansions of minimal weight f or β = 1 + √ 5 2 (left) and a compact representation of M β (right). It is of c ours e possible to follo w the proof of Theorem 3.10, bu t the states of A β are 0 , ± 1 β 3 , ± 1 β 2 , ± 1 β , ± 1 , ± β , ± β 2 , ± β ± 1 β 2 , ± β ± 1 β 3 , ± β 2 ± 1 β 2 , ± β 2 ± 1 β 3 , thus W = 2 and the transducer S β has 160 states. For other bases β , the number of states can b e mu ch larger . Therefore w e ha ve to re fine th e te chniques if w e d o not want computer -assis ted proofs. It is possible to show that a lar ge part of S β is not needed, e.g. by excluding some β -hea vy factors s uch as 11 from the output, and to obtain finally the transducer in Figure 2. Howe ver , it is easier to prove Theorem 4.1 by an indirect strategy , which includes some results which are interesting by themselves. Lemma 4.2. All words in {− 1 , 0 , 1 } ∗ which ar e not r ecognized by the autom aton M β in F igur e 1 are β -heavy . Pr oof. The transducer in Figure 2 is a part of the tr ansducer S β in the proof of Theo- rem 3.10. This means that eve r y word which is the input of a path (with full or dashed transitions) going from ( 0 , 0 ) to ( 0 , − 1 ) is β -hea vy , because the output has the s ame v alue but less weight. Since a β -hea vy word remains β -hea vy if we omit the leading and trailing zeros, the dashed transitions can be omitted. Then the set of inputs is H = 1 ( 0100 ) ∗ 1 ∪ 1 ( 0100 ) ∗ 0101 ∪ 1 ( 00 ¯ 10 ) ∗ ¯ 1 ∪ 1 ( 00 ¯ 10 ) ∗ 0 ¯ 1 ∪ ¯ 1 ( 0 ¯ 100 ) ∗ ¯ 1 ∪ ¯ 1 ( 0 ¯ 100 ) ∗ 0 ¯ 10 ¯ 1 ∪ ¯ 1 ( 0010 ) ∗ 1 ∪ ¯ 1 ( 0010 ) ∗ 01 and M β is constructed as in the proof of Lemma 3.9. ✷ Similarly to the N AF in base 2, where the expansions of minimal weight av oid the set { 11 , 1 ¯ 1 , ¯ 1 ¯ 1 , ¯ 11 } , we sho w in the next result that, for β = 1 + √ 5 2 , ev ery real number admits a β -expansion which av oids a certain finite s et X . Pr oposition 4.3. If β = 1 + √ 5 2 , then every z ∈ R has a β -e xpansion of the form z = y 1 · · · y k . y k + 1 y k + 2 · · · with y j ∈ {− 1 , 0 , 1 } such that y 1 y 2 · · · avoids the s et X = { 11 , 101 , 1001 , 1 ¯ 1 , 10 ¯ 1 , and their opposites } . If z ∈ Z [ β ] = Z [ β − 1 ] , then this e xpansion is unique up to leading zer os. 10 Christiane Frougny and W olfgang Steiner 0 , 0 − 1 , 1 − 1 /β , 0 0 , − 1 − 1 , 0 − 1 /β , − 1 − 1 , − 1 − 1 /β , − 2 1 , 1 1 /β , 0 0 , − 1 1 , 0 1 /β , − 1 1 , − 1 1 /β , − 2 1 | 0 1 | 0 0 | 0 1 | 0 0 | 0 1 | 0 0 | ¯ 1 ¯ 1 | 0 ¯ 1 | 0 ¯ 1 | 0 0 | 0 ¯ 1 | 0 0 | 0 ¯ 1 | 0 0 | 1 1 | 0 0 | 1 0 | ¯ 1 0 | ¯ 1 0 | 1 Figur e 2. Tran sducer with strictly β -hea vy words as inputs, β = 1 + √ 5 2 . Pr oof. W e determine this β -expansion similarly to the greedy β -expansion in the In- troduction. Note that the maximal valu e of . x 1 x 2 · · · for a sequence x 1 x 2 · · · av oiding the elements of X is . ( 1000 ) ω = β 2 / ( β 2 + 1 ) . If we define the transformation τ :  − β 2 β 2 + 1 , β 2 β 2 + 1  →  − β 2 β 2 + 1 , β 2 β 2 + 1  , τ ( z ) = β z −  β 2 + 1 2 β z + 1 / 2  , and set y j =  β 2 + 1 2 β τ j − 1 ( z ) + 1 / 2  for z ∈ h − β 2 β 2 + 1 , β 2 β 2 + 1  , j ≥ 1, then z = . y 1 y 2 · · · . I f y j = 1 for some j ≥ 1, then we hav e τ j ( z ) ∈ β ×  β β 2 + 1 , β 2 β 2 + 1  − 1 =  − 1 β 2 + 1 , 1 /β β 2 + 1  , hence y j + 1 = 0, y j + 2 = 0, and τ j + 2 ( z ) ∈  − β 2 β 2 + 1 , β β 2 + 1  , hence y j + 3 ∈ { ¯ 1 , 0 } . This show s that the given factors are av oided. A similar argument for y j = − 1 shows that the opposites a re a voided as w ell, hence we ha ve sho wn the e xistence of th e e xpansion for z ∈ h − β 2 β 2 + 1 , β 2 β 2 + 1  . For arbitrary z ∈ R , the exp ansion is gi ven by s hifting the expan sion of β − k z , k ≥ 0, to the left. If we choose y j = 0 in c as e τ j − 1 ( z ) > β / ( β 2 + 1 ) = . ( 0100 ) ω , then it is impossible to a v oid the factors 11, 101 and 1001 in the f ollo wing. If we choose y j = 1 in case τ j − 1 ( z ) < β / ( β 2 + 1 ) , then β τ j − 1 ( z ) − 1 < − 1 / ( β 2 + 1 ) = . ( 00 ¯ 10 ) ω , and thus it is impossible to a void the facto rs 1 ¯ 1, 10 ¯ 1, ¯ 1 ¯ 1, ¯ 10 ¯ 1 and ¯ 100 ¯ 1. Since β / ( β 2 + 1 ) 6∈ Z [ β ] , we have τ j − 1 ( z ) 6 = β / ( β 2 + 1 ) for z ∈ Z [ β ] . Similar relations hold f or the opposites, thus the ex pansion is unique. ✷ Remark 4.4. Similarly , the transformation τ ( z ) = β z − ⌊ z + 1 / 2 ⌋ on [ − β / 2 , β / 2 ) provides for every z ∈ Z [ β ] a unique expansion av oiding the factors 11, 101, 1 ¯ 1, 10 ¯ 1, 100 ¯ 1 and their opposites. Pr oposition 4.5. If x is accepted by M β , then there exists y ∈ {− 1 , 0 , 1 } ∗ avoiding X = { 11 , 101 , 1001 , 1 ¯ 1 , 10 ¯ 1 and their opposites } with x ∼ β y and k x k = k y k . The transducer N β in F igur e 3 realizes the c on version fr om 0 x 0 to y . Minimal weight expa nsions in Pisot bases 11 0 , 0; 0 0 | 0 0 , 0; 1 0 , 0; 10 0 , 0; 100 1 , 1 1 / β , 0 1 , 0 1 /β , 1 1 /β , − 1 − 1 /β 2 , 0 0 , 0; ¯ 1 0 , 0; ¯ 10 0 , 0; ¯ 100 − 1 , 1 − 1 /β , 0 − 1 , 0 − 1 /β , 1 − 1 /β , − 1 1 /β 2 , 0 1 | 1 0 | 0 0 | 0 0 | 0 ¯ 1 | ¯ 1 0 | ¯ 1 ¯ 1 | 0 0 | 0 0 | 1 ¯ 1 | 0 0 | 0 ¯ 1 | 0 ¯ 1 | 1 0 | 1 0 | 0 ¯ 1 | ¯ 1 0 | 0 0 | 0 0 | 0 1 | 1 0 | 1 1 | 0 0 | 0 0 | ¯ 1 1 | 0 0 | 0 1 | 0 1 | ¯ 1 0 | ¯ 1 0 | 0 0 | ¯ 1 0 | 1 Figur e 3. Tran sducer N β normalizing β -expansions of minimal weight, β = 1 + √ 5 2 . Pr oof. Set Q 0 = { ( 0 , 0; 0 ) , ( − 1 , 1 ) , ( 1 , 1 ) } = Q ′ 0 , Q 1 = { ( 0 , 0; 1 ) , ( − 1 /β , 0 ) } , Q ′ 1 = { ( 0 , 0; ¯ 10 ) } , Q 10 = { ( 0 , 0; 10 ) , ( − 1 , 0 ) } , Q ′ 10 = { ( 0 , 0; ¯ 100 ) } , Q 100 = { ( 0 , 0; 100 ) , ( − 1 /β , 1 ) } , Q ′ 100 = { ( 0 , 0; 0 ) , ( − 1 , 1 ) } , Q 101 = { ( − 1 /β , − 1 ) , ( 1 /β 2 , 0 ) } , Q ′ 101 = { ( 0 , 0; 1 ) } , and, symmetrically , Q ¯ 1 = { ( 0 , 0; ¯ 1 ) , ( 1 /β , 0 ) } , Q ′ ¯ 1 = { 0 , 0; 10 } , . . . . T hen the paths in N β with input in 00 ∗ lead to the three states in Q 0 , the paths with input 01 lead to the two states in Q 1 , and more generally the paths in N β with input 0 x such that x is accepted by M β lead to all states in Q u or to all states in Q ′ u , where u labels the shortest path in M β leading to the state reached by x . Indeed, if u a → v is a tr ansition in M β , then we ha ve Q u a → Q v or Q u a → Q ′ v , and Q ′ u a → Q v or Q ′ u a → Q ′ v , where Q a → R means that for ev ery r ∈ R there exists a transition q a | b − → r in N β with q ∈ Q . Since ev ery Q u and eve ry Q ′ u contains a s tate q with a transition of the form q 0 | b − → ( 0 , 0; w ) , there exists a path with input 0 x 0 going from ( 0 , 0; 0 ) to ( 0 , 0; w ) for ev ery word x accepted by M β . By construction, the output y of this path satisfies x ∼ β y and k x k = k y k . It can be easily checked that all outputs of N β a void the factors in X . ✷ Pr oof of Theor em 4.1. For eve ry x ∈ Z ∗ , by Proposition 3.1 and Lemma 4.2, there exists a β -expa ns ion of minimal weight y accepte d by M β with y ∼ β x . By Proposi- tion 4.5, there also exists a β -expansion of minimal weight y ′ ∈ { − 1 , 0 , 1 } ∗ a voiding X with y ′ ∼ β y ∼ β x . By Proposition 4.3, the output of N β is the same (if we neglect leading and trailing zeros) for ev ery input 0 x ′ 0 such that x ′ ∼ β x and x ′ is accepted by M β . Therefore k x ′ k = k y ′ k for all these x ′ , and the theorem is prov ed. ✷ 4.2 Branching transformation All β - expan sions of minimal weight can be obtained by a branching transformation. 12 Christiane Frougny and W olfgang Steiner Theor em 4.6. Let x = x 1 · · · x n ∈ {− 1 , 0 , 1 } ∗ and z = . x 1 · · · x n , β = 1 + √ 5 2 . Then x is a β - e xpansion of minimal weight if and only if − 2 β β 2 + 1 < z < 2 β β 2 + 1 and x j =                1 if 2 β 2 + 1 < β j − 1 z − x 1 · · · x j − 1 . < 2 β β 2 + 1 0 or 1 if β β 2 + 1 < β j − 1 z − x 1 · · · x j − 1 . < 2 β 2 + 1 0 if − β β 2 + 1 < β j − 1 z − x 1 · · · x j − 1 . < β β 2 + 1 − 1 or 0 if − 2 β 2 + 1 < β j − 1 z − x 1 · · · x j − 1 . < − β β 2 + 1 − 1 if − 2 β β 2 + 1 < β j − 1 z − x 1 · · · x j − 1 . < − 2 β 2 + 1 for all j, 1 ≤ j ≤ n. The sequence ( β j − 1 z − x 1 · · · x j − 1 . ) 1 ≤ j ≤ n is a trajectory ( τ j − 1 ( z )) 1 ≤ j ≤ n , where the branching transformation τ : z 7→ β z − x 1 with x 1 ∈ {− 1 , 0 , 1 } is gi ven in Figure 4.  − 2 β β 2 +1 , − β β 2 +1   − 2 β 2 +1 , − 2 β β 2 +1   − β β 2 +1 , 1 β 2 +1   β β 2 +1 , − 1 β 2 +1   2 β 2 +1 , 2 β β 2 +1   2 β β 2 +1 , β β 2 +1  0 Figur e 4. Branching transformation giving all 1 + √ 5 2 -expansions of minimal weight. Pr oof. T o s ee that all words x 1 · · · x n gi ven by the branching transformation are β - expan sions of minimal weight, we hav e drawn in Figure 5 an automa ton where ev ery state is la beled by the interval containing all numbers β j z − x 1 · · · x j . such that x 1 · · · x j labels a path leading to this state. This automaton turns out to be the automaton M β in Figure 1 (up to the labels of the states), which accepts exactly the β -expa ns ions of minimal weight. Recall that . ( 0010 ) ω = 1 β 2 + 1 and thus . 1 ( 0100 ) ω = 2 β β 2 + 1 . If the conditions on z and x j are not s atisfied, then we hav e either | . x j · · · x n | > . 1 ( 0100 ) ω , or x j = 1 and . x j + 1 · · · x n < . ( 00 ¯ 10 ) ω , or x j = − 1 and . x j + 1 · · · x n > . ( 0010 ) ω for some j , 1 ≤ j ≤ n . In ev ery case, it is easy to see that x j · · · x n must contain a f actor in the set H of the proof of Lemma 4.2, hence x 1 · · · x n is β -hea vy . ✷ 4.3 Fibonacci numeration system The reader is referred to [18, Chapter 7] for d efinitions on numeration systems de fined by a sequence of integers. Recall that the linear numeration system canonically as- sociated with the Golden Ratio is the Fibonacci (or Zeckendorf) numeration s ystem Minimal weight expa nsions in Pisot bases 13 . ¯ 1(0 ¯ 100) ω , . 1(0100) ω . (00 ¯ 10) ω , . (0100) ω . (0 ¯ 100) ω , . (1000) ω . ( ¯ 1000) ω , . 1(0100) ω . (000 ¯ 1) ω , . (0010) ω . (0 ¯ 100) ω , . (0010) ω . ( ¯ 1000) ω , . (0100) ω . ¯ 1(0 ¯ 100) ω , . (1000) ω . (00 ¯ 10) ω , . (0001) ω 0 1 0 0 1 ¯ 1 0 1 ¯ 1 0 0 ¯ 1 1 0 ¯ 1 0 0 Figur e 5. Automaton M β with interv als as labels. defined by the sequence of Fibona cci numbers F = ( F n ) n ≥ 0 with F n = F n − 1 + F n − 2 , F 0 = 1 and F 1 = 2. Any non-negati ve integer N < F n can be represented as N = P n j = 1 x j F n − j with the property that x 1 · · · x n ∈ { 0 , 1 } ∗ does not contain the factor 11. For words x = x 1 · · · x n ∈ Z ∗ , y = y 1 · · · y m ∈ Z ∗ , we define a relation x ∼ F y if and only if n X j = 1 x j F n − j = m X j = 1 y j F m − j . The properties F -heavy and F -expansion of minimal weight are defined as for β - expan sions, with ∼ F instead of ∼ β . An important difference between the notions F -hea vy and β -hea vy is that a wo r d containing a F -heavy fac tor need not be F -heavy , e.g. 2 is F -heavy since 2 ∼ F 10, b ut 20 is not F -heavy . Ho weve r , uxv is F -hea vy if x 0 length ( v ) is F -hea vy . T herefore we s ay that x ∈ Z ∗ is str ongly F -heavy if e very element in x 0 ∗ is F -heavy . Hence ev ery word containing a strongly F -hea vy fa ctor is F -hea vy . The Golden Ratio satisfies (D 2 ) since 2 = 10 . 01. F or the Fibonacci numbers, the corresponding relation is 2 F n = F n + 1 + F n − 2 , hence 20 n ∼ F 10010 n − 2 for all n ≥ 2. Since 20 ∼ F 101 and 2 ∼ F 10, we obtain similarly to the proof of Proposition 3.1 that for ev ery x ∈ Z ∗ there exists some y ∈ {− 1 , 0 , 1 } ∗ with x ∼ F y and k y k ≤ k x k . W e will show the follow ing theorem. Theor em 4.7. The set of F -expansions of minimal w eight in {− 1 , 0 , 1 } ∗ is equal to the set of β -expansions of minimal weight in {− 1 , 0 , 1 } ∗ for β = √ 5 + 1 2 . The proof of this theorem runs along the same lines as the proof of Theorem 4.1. W e use the unique expansion of integers gi ven by Proposition 4. 8 (due to Heuberge r [15]) and provide an alternativ e proof of Heuber ger’ s result that these expansions are F - expan sions of minimal weight. Pr oposition 4.8 ([15]). Every N ∈ Z has a unique repr esentation N = P n j = 1 y j F n − j with y 1 6 = 0 and y 1 · · · y n ∈ { − 1 , 0 , 1 } ∗ avoiding X = { 11 , 101 , 1001 , 1 ¯ 1 , 10 ¯ 1 , and their opposites } . Pr oof. Let g n be the smallest positi ve inte ger with a n F -expansion of le ngth n starting with 1 and a v oiding X , and G n be the largest integer of this kind. Since g n + 1 ∼ F 14 Christiane Frougny and W olfgang Steiner 1 ( 00 ¯ 10 ) n/ 4 , G n ∼ F ( 1000 ) n/ 4 and 1 ( ¯ 10 ¯ 10 ) n/ 4 ∼ F 1, we obtain g n + 1 − G n = 1. (A fractional power ( y 1 · · · y k ) j /k denotes the word ( y 1 · · · y k ) ⌊ j /k ⌋ y 1 · · · y j −⌊ j /k ⌋ k .) Therefore th e length n of an e xpansion y 1 y 2 · · · y n of N 6 = 0 with y 1 6 = 0 a voiding X is determined by G n − 1 < | N | ≤ G n . Since g n − F n − 1 = − G n − 3 and G n − F n − 1 = G n − 4 , we have − G n − 3 ≤ N − F n − 1 ≤ G n − 4 if y 1 = 1, hence y 2 = y 3 = 0, y 4 6 = 1, and we obtain recursi vely that N has a unique expa ns ion av oiding X . ✷ 0 , 0 − 1 , 1 − 1 /β , 0 0 , − 1 − 1 , 0 − 1 /β , − 1 − 1 , − 1 − 1 /β , − 2 1 /β 2 , − 1 1 , 1 1 /β , 0 0 , − 1 1 , 0 1 /β , − 1 1 , − 1 1 /β , − 2 − 1 /β 2 , − 1 1 | 0 1 | 0 0 | 0 1 | 0 0 | 0 0 | ¯ 1 1 | 0 ¯ 1 | 0 ¯ 1 | 0 ¯ 1 | 0 0 | 0 ¯ 1 | 0 0 | 0 ¯ 1 | 0 0 | 1 1 | 0 0 | 0 0 | ¯ 1 1 | ¯ 1 0 | 0 0 | 1 ¯ 1 | 1 0 | ¯ 1 0 | 1 Figur e 6. All inputs of this transducer are strongly F -hea vy . Pr oof of Theor em 4.7. Let a 1 · · · a n ∈ Z ∗ , z = P n j = 1 a j β n − j , N = P n j = 1 a j F n − j . By using the equations β k = β k − 1 + β k − 2 and F k = F k − 1 + F k − 2 , we obtain integers m 0 and m 1 such that z = m 1 β + m 0 and N = m 1 F 1 + m 0 F 0 = 2 m 1 + m 0 . Clearly , z = 0 implies m 1 = m 0 = 0 and thus N = 0, but the con verse is not true: N = 0 only implies m 0 = − 2 m 1 , i.e., z = − m 1 /β 2 . Therefore we hav e x 1 · · · x n ∼ F y 1 · · · y n if and only if ( x 1 − y 1 ) · · · ( x n − y n ) . = m/ β 2 for some m ∈ Z , hence the redundanc y transducer R F for the Fibonacci numeration system is similar to R β , except that all states m/β 2 , m ∈ Z , are terminal. The transducer in Figure 6 s ho ws that all strictly β -heavy words in {− 1 , 0 , 1 } ∗ are strongly F -heavy . Therefore all words which are not accepted by M β are F -heavy . Let N F be as N β , except that the states ( ± 1 /β 2 , 0 ) are terminal. Every set Q u and Q ′ u contains a state of the form ( 0 , 0; w ) or ( ± 1 /β 2 , 0 ) . If x is accepted by N β , then N F transforms therefore 0 x into a word y av oiding the factors giv en in Proposition 4.8. Hence x is an F -expa ns ion of minimal weight. ✷ Remark 4.9. If we consider only expansions av oiding the factors 11, 101, 1 ¯ 1, 10 ¯ 1, 100 ¯ 1, then the difference between the lar gest integer with expansion of length n and the smallest positiv e integer with expansion of length n + 1 is 2 if n is a positiv e multiple of 3. Therefore there exist integers without an expansion of this kind, e.g. N = 4. Ho weve r, a small modification pro vides anoth er “nice” set of F -expansions of minimal weight: Every integer has a uniqu e representation of the form N = P n j = 1 y j F n − j with y 1 6 = 0, y 1 · · · y n ∈ { ¯ 1 , 0 , 1 } ∗ a voiding the factors 11 , ¯ 1 ¯ 1 , ¯ 10 ¯ 1 , 1 ¯ 1 , ¯ 11 , 10 ¯ 1 , ¯ 101 , 100 ¯ 1 and y j − 2 y j − 1 y j = 101 or y j − 3 · · · y j = ¯ 1001 only if j = n . Minimal weight expa nsions in Pisot bases 15 4.4 W eight of the expan sions In this section, we study the a verage weight of F -expansions of minimal weight. For e very N ∈ Z , let k N k F be the weight of a corresponding F -expansion of minimal weight, i.e., k N k F = k x k if x is an F -expansion of minimal weight with x ∼ F N . Theor em 4. 10. F or positive inte gers M , we have , as M → ∞ , 1 2 M + 1 M X N = − M k N k F = 1 5 log M log 1 + √ 5 2 + O ( 1 ) . Pr oof. Consider first M = G n for some n > 0, where G n is defined as in the proof of Proposition 4.8, and let W n be the set of words x = x 1 · · · x n ∈ { − 1 , 0 , 1 } n a voiding 11 , 101 , 1001 , 1 ¯ 1 , 10 ¯ 1, and their opposites. Then we hav e 1 2 G n + 1 G n X N = − G n k N k F = 1 # W n X x ∈ W n k x k = n X j = 1 E X j , where E X j is the expe cted value of the random v ariable X j defined by Pr [ X j = 1 ] = # { x 1 · · · x n ∈ W n : x j 6 = 0 } # W n , Pr [ X j = 0 ] = # { x 1 · · · x n ∈ W n : x j = 0 } # W n Instead of ( X j ) 1 ≤ j ≤ n , we consider the sequence of random variables ( Y j ) 1 ≤ j ≤ n de- fined by Pr [ Y 1 = y 1 y 2 y 3 , . . . , Y j = y j y j + 1 y j + 2 ] = # { x 1 · · · x n + 2 ∈ W n 00 : x 1 · · · x j + 2 = y 1 · · · y j + 2 } / # W n , Pr [ Y j − 1 = xy z , Y j = x ′ y ′ z ′ ] = 0 if x ′ 6 = y or y ′ 6 = z . It is easy to see that ( Y j ) 1 ≤ j ≤ n is a Marko v chain, where the non-tri vial transition probabilities are given by 1 − Pr [ Y j + 1 = 000 | Y j = 100 ] = Pr [ Y j + 1 = 00 ¯ 1 | Y j = 100 ] = G n − j − 2 − G n − j − 3 G n − j + 1 − G n − j , 1 − 2 Pr [ Y j + 1 = 001 | Y j = 000 ] = Pr [ Y j + 1 = 000 | Y j = 000 ] = 2 G n − j − 3 + 1 2 G n − j − 2 + 1 , and the opposite relations. Since G n = cβ n + O ( 1 ) (with β = 1 + √ 5 2 , c = β 3 / 5), the transition probabilities satisfy Pr [ Y j + 1 = v | Y j = u ] = p u,v + O ( β − n + j ) with ( p u,v ) u,v ∈ { 100 , 010 , 001 , 000 , 00 ¯ 1 , 0 ¯ 10 , 00 ¯ 1 } =              0 0 0 2 β 2 1 β 3 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 2 β 2 1 β 1 2 β 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 β 3 2 β 2 0 0 0              . 16 Christiane Frougny and W olfgang Steiner The eigen values of this matrix are 1 , − 1 β , ± i β , 1 ± i √ 3 2 β , − 1 β 2 . The s tationary dis tributio n vector (given by the left eigen vector to the eigen value 1) is ( 1 10 , 1 10 , 1 10 , 2 5 , 1 10 , 1 10 , 1 10 ) , thus we ha ve E X j = Pr [ Y j = 100 ] + Pr [ Y j = ¯ 100 ] = 1 / 5 + O  β − min ( j,n − j )  , cf. [8]. This prov es the theorem for M = G n . If G n < M ≤ G n + 1 , then we hav e k N k F = 1 + k N − F n k F if G n < N ≤ M , and a similar relation for − M ≤ N < − G n . W ith G n + 1 − F n = − G n − 2 , we obtain M X N = − M k N k F = G n X N = − G n k N k F + M − F n X N = − G n − 2 ( 1 + k N k F ) + G n − 2 X N = F n − M ( 1 + k N k F ) = G n X N = − G n k N k F + G n − 2 X N = − G n − 2 k N k F + sgn ( M − F n ) | M − F n | X N = −| M − F n | k N k F + O ( M ) = 2 5 log β  F n log M + ( M − F n ) log | M − F n |  + O ( M ) = 2 M log M 5 log β + O ( M ) by induction on n and using M − F n M log | M − F n M | = O ( 1 ) . ✷ Remark 4.11. As in [8], a central limit theorem for the distribution of k N k F can be prov ed, ev en if we restrict the numbers N to polynomial sequences or prime numbers. Remark 4.12. If we partition the interv al  − β 2 β 2 + 1 , β 2 β 2 + 1  , where the transformation τ : z 7→ β z −  β 2 + 1 2 β z + 1 / 2  of the proof of Proposition 4.3 is defined, into intervals I ¯ 100 =  − β 2 β 2 + 1 , − β β 2 + 1  , I 0 ¯ 10 =  − β β 2 + 1 , − 1 β 2 + 1  , I 00 ¯ 1 =  − 1 β 2 + 1 , − 1 /β β 2 + 1  , I 000 =  − 1 /β β 2 + 1 , 1 /β β 2 + 1  , I 001 =  1 /β β 2 + 1 , 1 β 2 + 1  , I 010 =  1 β 2 + 1 , β β 2 + 1  , I 100 =  β β 2 + 1 , β 2 β 2 + 1  , then we ha ve p u,v = λ ( τ ( I u ) ∩ I v ) /λ ( τ ( I u )) , where λ denotes the Lebesgue measure. 5 T ri bonacci case In this section, let β > 1 be the Tribonac ci number, β 3 = β 2 + β + 1 ( β ≈ 1 . 839). Since 1 = . 111, we hav e 2 = 10 . 001 and β satisfies (D 2 ). Here, the digits of arbitrary β - expa nsions of minimal weight are in {− 5 , . . . , 5 } since 6 = 1000 . 00 ¯ 10 ¯ 10 ¯ 1. W e h a ve 5 = 101 . 100011 and we will s ho w th at 101100011 is a β -expansion of minimal weight, thus 5 is also a β -expansion of minimal weight. The proofs of the results in this section run along the same lines as in the Golden Ratio case. Therefore we gi ve only an outline of them. 5.1 β -expansio ns of minimal weight All words which are not accepted by the automaton M β in Figure 7, where all states are terminal, are β - hea vy since they contain a factor which is accepted by the input automaton of the transducer in Figure 8 (without the dashed arro ws). Minimal weight expa nsions in Pisot bases 17 0 1 ¯ 1 1 0 ¯ 1 0 0 0 1 0 0 ¯ 1 0 1 0 1 ¯ 1 ¯ 1 1 ¯ 1 0 1 0 0 0 ¯ 1 0 0 1 0 ¯ 1 0 ¯ 1 1 0 0 Figur e 7. Automata M β , β 3 = β 2 + β + 1, and M T . Pr oposition 5.1. If β > 1 is the T ribonacci number , then every z ∈ R has a β - e xpansion of the for m z = y 1 · · · y k . y k + 1 y k + 2 · · · with y j ∈ {− 1 , 0 , 1 } such that y 1 y 2 · · · avoids the set X = { 11 , 101 , 1 ¯ 1 , and their opposites } . If z ∈ Z [ β ] = Z [ β − 1 ] , then this e xpansion is unique up to leading zer os. The expansion in Proposition 5.1 is gi ven by the transformation τ :  − β β + 1 , β β + 1  →  − β β + 1 , β β + 1  , τ ( z ) = β z −  β + 1 2 z + 1 2  . Note that the wo rd av oiding X with maximal va lue is ( 100 ) ω , . ( 100 ) ω = β β + 1 . Remark 5.2. The transformation τ ( z ) = β z −  β 2 − 1 2 z + 1 2  on  − β β 2 − 1 , β β 2 − 1  provides a unique exp ansion av oiding the factors 11, 1 ¯ 1, 10 ¯ 1 and their opposites. Pr oposition 5.3. The con version of an arbitrary e xpansion acce pted by the autom aton M β in F igure 7 into the exp ansion avoiding X = { 11 , 101 , 1 ¯ 1 , and their opposites } is r ealized by the transducer N β in F igur e 9 and does not cha nge the weight. Theor em 5.4. If β is the T ribonacci number , then the set of β -e xpansions of minimal weight in {− 1 , 0 , 1 } ∗ is reco gnized by the finite automaton M β of F igure 7 wher e all states ar e terminal. 5.2 Branching transformation Contrary to the Golden Ratio case, we cannot obtain all β -expansions of minimal weight by the help of a piece wise linear branch ing transformation: If z = . 01 ( 001 ) n , then we ha ve no β -expansion of minimal weight of the f orm z = . 1 x 2 x 3 · · · , whereas z ′ = . 0011 has the e xpansion . 1 ¯ 1, and z ′ < z . On the other hand, z = . 1 ( 100 ) n 11 has no β -expansion of minimal weight of the form z = . 1 x 2 x 3 · · · (s ince 1 ( 100 ) n 11 is β - hea vy b ut ( 100 ) n 11 is not β -hea vy), whereas z ′ = . 1101 is a β -expa nsion of minimal 18 Christiane Frougny and W olfgang Steiner 0 , 0 − 1 , 1 1 − β, 0 − 1 /β , − 1 − 1 , − 1 1 − β , − 2 − 1 /β , − 3 1 − 1 /β , − 2 1 /β 3 , − 1 1 /β 2 , − 1 1 /β − 1 , − 2 1 /β 2 − 1 , − 2 − 1 − 1 /β 2 , − 2 1 /β 3 − 1 /β , − 2 1 /β 3 , − 3 − 1 /β 2 , − 3 0 , − 2 0 , 0 1 , 1 β − 1 , 0 1 /β , − 1 1 , − 1 β − 1 , − 2 1 /β , − 3 1 /β − 1 , − 2 − 1 /β 3 , − 1 − 1 /β 2 , − 1 1 − 1 /β , − 2 1 − 1 /β 2 , − 2 1 + 1 /β 2 , − 2 1 /β − 1 /β 3 , − 2 − 1 /β 3 , − 3 1 /β 2 , − 3 ¯ 1 | 0 1 | 0 1 | 0 1 | 0 0 | 0 0 | ¯ 1 1 | 0 0 | ¯ 1 1 | 0 0 | 0 ¯ 1 | 0 0 | 0 0 | ¯ 1 ¯ 1 | 0 1 | 0 0 | 0 1 | ¯ 1 0 | 0 1 | 0 1 | 0 ¯ 1 | 0 ¯ 1 | 0 ¯ 1 | 0 0 | 0 0 | 1 ¯ 1 | 0 0 | 1 ¯ 1 | 0 0 | 0 1 | 0 0 | 0 0 | 1 1 | 0 ¯ 1 | 0 0 | 0 ¯ 1 | 1 0 | 0 ¯ 1 | 0 0 | 0 1 | ¯ 1 0 | 0 ¯ 1 | 1 0 | 0 0 | 1 0 | ¯ 1 1 | ¯ 1 0 | ¯ 1 0 | 0 0 | ¯ 1 0 | 0 ¯ 1 | 1 0 | 1 0 | 1 0 | 0 0 | 0 Figur e 8. The relev ant part of S β , β 3 = β 2 + β + 1, and S T . weight, and z ′ > z . Hence the maximal interv al for the digit 1 is [ . ( 010 ) ω , . 1 ( 100 ) ω ] , with . ( 010 ) ω = β β 3 − 1 = 1 β + 1 and . 1 ( 100 ) ω = 2 β + 1 β ( β + 1 ) . The corresponding br anching transformation and the possible expan sions are giv en in Figure 10. 5.3 T ribonacci numerat ion system The linear numeration system canonically associated with the T ribonacci number is the T ribonacci numeration system defined by the sequence T = ( T n ) n ≥ 0 with T 0 = 1, T 1 = 2, T 2 = 4, and T n = T n − 1 + T n − 2 + T n − 3 for n ≥ 3. Any non-negati ve integer N < T n has a representation N = P n j = 1 x j T n − j with the property that x 1 · · · x n ∈ { 0 , 1 } ∗ does not contain the factor 111. The r elation ∼ T and the properties T -heavy , T -exp ansion of minimal weight and str ongly T -heavy are defined analogously to the Fibonacci numeration system. W e hav e 20 n ∼ T 100010 n − 3 for n ≥ 3, 200 ∼ T 1001, Minimal weight expa nsions in Pisot bases 19 0 , 0; 0 0 , 0; ¯ 1 0 , 0; ¯ 10 − 1 , 1 1 − β , 0; 0 − 1 /β , 1 − 1 /β , − 1; 0 − 1 , − 1 − 1 − 1 /β , 0 1 − β , 0; ¯ 1 − 1 /β, − 1; 1 − 1 /β 2 , − 1; 0 − 1 /β 2 , − 1; 10 1 /β 2 − 1 , − 2 − 1 /β 3 , − 1; 0 1 − 1 /β , 0 1 − 1 /β , − 2 − 1 /β 3 , − 1; 1 − 1 /β 2 , − 1; ¯ 1 − 1 /β 2 , − 1; 1 0 , 0; 1 0 , 0; 10 1 , 1 β − 1 , 0; 0 1 /β , 1 1 /β , − 1; 0 1 , − 1 1 + 1 /β , 0 β − 1 , 0; 1 1 /β , − 1; ¯ 1 1 /β 2 , − 1; 0 1 /β 2 , − 1; ¯ 10 1 − 1 /β 2 , − 2 1 /β 3 , − 1; 0 1 /β − 1 , 0 1 /β − 1 , − 2 1 /β 3 , − 1; ¯ 1 1 /β 2 , − 1; 1 1 /β 2 , − 1; ¯ 1 0 | 1 0 | ¯ 1 ¯ 1 | 0 1 | 0 1 | 0 ¯ 1 | 0 0 | ¯ 1 0 | 1 1 | 0 ¯ 1 | 0 1 | ¯ 1 ¯ 1 | 1 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 0 | 0 0 | 0 1 | ¯ 1 ¯ 1 | 1 1 | 0 ¯ 1 | 0 0 | ¯ 1 0 | 1 1 | 1 ¯ 1 | ¯ 1 0 | 0 0 | 0 0 | ¯ 1 0 | 1 1 | 0 ¯ 1 | 0 0 | ¯ 1 0 | 1 0 | 0 0 | 0 1 | 1 ¯ 1 | ¯ 1 ¯ 1 | 0 1 | 0 0 | 1 0 | ¯ 1 0 | 0 0 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 ¯ 1 | 0 1 | 0 0 | 0 0 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 ¯ 1 | ¯ 1 1 | 1 0 | ¯ 1 0 | 1 1 | ¯ 1 ¯ 1 | 1 0 | 0 0 | 0 0 | 0 0 | 0 0 | ¯ 1 0 | 1 0 | 1 0 | ¯ 1 0 | 0 Figur e 9. Normalizing transducer N β , β 3 = β 2 + β + 1. 20 ∼ T 100 and 2 ∼ T 10, therefore for every x ∈ Z ∗ there exists some y ∈ { − 1 , 0 , 1 } ∗ with x ∼ T y and k y k ≤ k x k . Since the differenc e of 1 ( 0 ¯ 10 ) n/ 3 and ( 100 ) n/ 3 is 1 ( ¯ 1 ¯ 10 ) n/ 3 ∼ T 1, we obtain the follo wing proposition. Pr oposition 5.5. Every N ∈ Z has a unique r epresenta tion N = P n j = 1 y j T n − j with y 1 6 = 0 and y 1 · · · y n ∈ {− 1 , 0 , 1 } ∗ avoiding X = { 11 , 101 , 1 ¯ 1 , and their opposites } . If z = a 1 · · · a n . = m 2 m 1 m 0 . , then N = P n j = 1 a j T n − j = 4 m 2 + 2 m 1 + m 0 = 0 if and only if m 0 = 2 m ′ 0 and m 1 = − 2 m 2 − m ′ 0 , i.e., z = − m 2 /β 2 + m ′ 0 /β 3 , hence all s tates s = m/ β 2 + m ′ /β 3 with some m, m ′ ∈ Z are terminal s tates in the redundanc y transducer R T . The transducer S T , which is given by Figure 8 includ- ing the dashed arrows exce pt that the states ( ± 1 /β , − 3 ) are not terminal, shows that all str ictly β -heavy words in {− 1 , 0 , 1 } ∗ are strongly T - hea vy , but that some other x ∈ {− 1 , 0 , 1 } ∗ are T -heavy as well. Thus the T -expansions of minimal weight are a subset of the set recognized by the automaton M β in Figure 7. Every set Q u and Q ′ u , u ∈ { 0 , 1 , 10 , 11 } , contains a terminal state ( 0 , 0; w ) or ( 1 − 1 / β , 0 ) , hence the w ords labelling paths ending in these states are T -expansions of minimal weight. The sets Q u and Q ′ u , u ∈ { 1 ¯ 1 , 1 ¯ 10 , 1 ¯ 11 , 1 ¯ 10 , 1 ¯ 101 } , contain states ( ± 1 / β 3 , − 1; w ) , 20 Christiane Frougny and W olfgang Steiner  − 2 β − 1 β ( β +1) , − β β + 1   − 2 − 1 /β β ( β +1) , − 2 β − 1 β ( β +1)   − 1 β + 1 , 1 β + 1   1 β + 1 , − 1 β + 1   2+1 /β β ( β +1) , 2 β +1 β ( β +1)   2 β +1 β ( β +1) , β β + 1  0 . ¯ 1( ¯ 100) ω , . 1(100) ω . (0 ¯ 10) ω , . (100) ω . (0 ¯ 10) ω , . (001) ω . ( ¯ 100) ω , . 1(100) ω . ( ¯ 100) ω , . (010) ω . (00 ¯ 1) ω , . (010) ω . ¯ 1( ¯ 100) ω , . (100) ω 0 1 0 1 0 1 ¯ 1 ¯ 1 0 ¯ 1 0 ¯ 1 1 0 0 Figur e 10. Branching transformation, corresponding automaton, β 3 = β 2 + β + 1. ( ± 1 /β 2 , − 1; w ) , ( ± ( 1 − 1 /β ) , − 2 ) , hence the words labelling paths ending in these states are T -hea vy , and we obtain the followin g theorem. Theor em 5.6. The T - e xpansions of minimal weight in {− 1 , 0 , 1 } ∗ ar e exac tly the wor ds which are accepted by M T , w hich is the automaton in F igure 7 where only the states with a dashed outgoing arr ow ar e terminal. The words given by Pr oposition 5.5 ar e T -expansions o f minimal weight. 5.4 W eight of the expan sions Let W n be the set of words x = x 1 · · · x n ∈ { − 1 , 0 , 1 } n a voiding the factors 11, 101, 1 ¯ 1, and their opposites. Then the sequence of random va r iables ( Y j ) 1 ≤ j ≤ n defined by Pr [ Y 1 = y 1 y 2 , . . . , Y j = y j y j + 1 ] = # { x 1 · · · x n + 1 ∈ W n 0 : x 1 · · · x j + 1 = y 1 · · · y j + 1 } # W n is Marko v with transition probabilities Pr [ Y j + 1 = v | Y j = u ] = p u,v + O ( β − n + j ) , ( p u,v ) u,v ∈ { 10 , 01 , 00 , 0 ¯ 1 , ¯ 10 } =         0 0 β 2 − 1 β 2 1 β 2 0 1 0 0 0 0 0 β − 1 2 β 1 β β − 1 2 β 0 0 0 0 0 1 0 1 β 2 β 2 − 1 β 2 0 0         . The eigen values of this matrix are 1 , ± 1 β , − β − 1 ± i √ 3 β 3 − β 2 β 3 , and the stationary distr i- b ution vector of the Markov chain is  β 3 / 2 β 5 + 1 , β 3 / 2 β 5 + 1 , β 3 + β 2 β 5 + 1 , β 3 / 2 β 5 + 1 , β 3 / 2 β 5 + 1  . W e obtain the follo wing theorem (with β 3 β 5 + 1 = . ( 0011010100 ) ω ≈ 0 . 28219). Minimal weight expa nsions in Pisot bases 21 Theor em 5. 7. F or positive inte gers M , we have , as M → ∞ , 1 2 M + 1 M X N = − M k N k T = β 3 β 5 + 1 log M log β + O ( 1 ) . 6 Smallest Pisot number case The smallest Pis ot number β ≈ 1 . 325 s atisfies β 3 = β + 1. Since 1 = . 011 = . 10001 implies 2 = 100 . 00001 as well as 2 = 1000 . 000 ¯ 1, (D 2 ) holds. W e hav e furthermore 3 = β 4 − β − 9 , thus all β -expansions of minimal weight hav e digits in {− 2 , . . . , 2 } . 6.1 β -expansio ns of minimal weight 1 ¯ 1 ¯ 1 1 1 ¯ 1 ¯ 1 ¯ 1 ¯ 1 1 1 ¯ 1 0 0 0 0 1 ¯ 1 1 ¯ 1 0 0 0 0 0 0 0 0 1 ¯ 1 0 0 ¯ 1 1 0 0 1 ¯ 1 0 0 0 0 0 0 0 0 0 0 0 0 ¯ 1 1 1 ¯ 1 0 0 1 ¯ 1 ¯ 1 1 0 0 0 0 1 ¯ 1 0 0 0 0 0 0 0 0 ¯ 1 1 0 Figur e 11. Automata M β , β 3 = β + 1, and M S . Let M β be the automaton in Figure 11 without the dashed arro ws where all states are terminal. Then it is a bit more difficult than in the Golden Ration and th e T ribonacci cases to see that all words which are not accepted by M β are β -hea vy , not only because the automa ta are large r but also beca use some inputs of the transducer in Figure 13 a re not strictly β -heavy (bu t of course still β -hea vy). W e r efer to [13] for details. Pr oposition 6.1. If β is th e smalle s t Pisot num ber , then e very z ∈ R has a β -ex pansion of the form z = y 1 · · · y k . y k + 1 y k + 2 · · · with y j ∈ {− 1 , 0 , 1 } such that y 1 y 2 · · · avoids the 22 Christiane Frougny and W olfgang Steiner 0 ¯ 10 7 10 7 10 7 ¯ 10 7 0 2 0 2 0 4 ¯ 10 2 , 0 2 10 4 , ¯ 10 7 0 4 10 2 , 0 2 ¯ 10 4 , 10 7 0 4 10 0 4 ¯ 10 0 3 10 2 , 0 3 ¯ 10 3 0 3 ¯ 10 2 , 0 3 10 3 0 0 0 2 ¯ 10 3 0 2 10 3 010 3 0 ¯ 10 3 010 4 0 ¯ 10 4 0 2 0 2 10 6 ¯ 10 6 0 5 0 5 10 ¯ 10 Figur e 12. C ompact representation of M β . set X = { 10 6 1 , 10 k 1 , 10 k ¯ 1 , 0 ≤ k ≤ 5 , and their opposites } . If z ∈ Z [ β ] = Z [ β − 1 ] , then this e xpansion is unique up to leading zer os. The expansion in Proposition 6.1 is gi ven by the transformation τ : h − β 3 β 2 + 1 , β 3 β 2 + 1  → h − β 3 β 2 + 1 , β 3 β 2 + 1  , τ ( x ) = β x − j β 2 + 1 2 β 2 x + 1 2 k since τ  β 2 β 2 + 1 , β 3 β 2 + 1  =  β 3 β 2 + 1 − 1 , β 4 β 2 + 1 − 1  =  − 1 /β 3 β 2 + 1 , 1 /β 4 β 2 + 1  . The word av oiding X with maximal va lue is ( 10 7 ) ω , . ( 10 7 ) ω = β 7 / ( β 8 − 1 ) = β 3 / ( β 2 + 1 ) . Remark 6.2. The transformation τ ( z ) = β z −  1 β z + 1 2  on  − β 2 2 , β 2 2  provides a unique ex pansion av oiding 10 6 ¯ 1 instead of 10 6 1. Pr oposition 6.3. The con version of an arbitrary e xpansion accepted by M β into the e xpansion avoiding X = { 10 6 1 , 10 k 1 , 10 k ¯ 1 , 0 ≤ k ≤ 5 , and their opposites } is re al- ized by the transduce r N β in F igur e 14 and does not ch ange the weight. Theor em 6 .4. If β is the s mallest Pisot number , then the set of β - e xpansions o f minimal weight in {− 1 , 0 , 1 } ∗ is rec ognized by the finite automaton M β of F igure 11 (without the dashed arr ows) wher e all states ar e terminal. 6.2 Branching transformation In the case of the smallest Pisot number β , the maximal interv al f or the digit 1 is [ . ( 010 6 ) ω , . 1 ( 0 5 10 2 ) ω ] , with . ( 010 6 ) ω = β 2 β 2 + 1 and . 1 ( 0 5 10 2 ) ω = β 2 + 1 /β β 2 + 1 . The corre- sponding branching transformation and expa ns ions are giv en in Figure 15. Minimal weight expa nsions in Pisot bases 23 0 , 0 0 , − 1 0 , − 2 − 1 /β 5 , 0 1 /β 5 , 0 − 1 /β 4 , 0 1 /β 4 , 0 − 1 /β 3 , 0 1 /β 3 , 0 − 1 /β 2 , 0 1 /β 2 , 0 − 1 /β , 0 1 /β , 0 − 1 , 0 1 , 0 − β, 0 β, 0 − β 2 , 0 β 2 , 0 − 1 /β 5 , − 1 1 /β 5 , − 1 − 1 /β 4 , − 1 1 /β 4 , − 1 − 1 /β 3 , − 1 1 /β 3 , − 1 − 1 /β 2 , − 1 1 /β 2 , − 1 − 1 /β , − 1 1 /β , − 1 − 1 , − 1 1 , − 1 − β, − 1 β, − 1 − 1 /β 5 , − 2 1 /β 5 , − 2 − 1 /β 4 , − 2 1 /β 4 , − 2 − 1 /β 3 , − 2 1 /β 3 , − 2 − 1 /β 2 , − 2 1 /β 2 , − 2 − 1 /β , − 2 1 /β , − 2 − 1 , 1 1 , 1 − β, 1 β, 1 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 0 | ¯ 1 0 | 1 0 | ¯ 1 0 | 1 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 ¯ 1 | 0 1 | 0 0 | 1 0 | ¯ 1 0 | 0 0 | 0 0 | ¯ 1 0 | 1 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 0 | ¯ 1 0 | 1 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 1 | 0 1 | 0 Figur e 13. The relev ant part of S β , β 3 = β + 1. 6.3 Integer expansions Let ( S n ) n ≥ 0 be a linear numeration system ass ociated with the smallest Pisot number β which is define d as f ollo ws: S 0 = 1 , S 1 = 2 , S 2 = 3 , S 3 = 4 , S n = S n − 2 + S n − 3 for n ≥ 4 . Note that we do not choose the cano nical nume ration system ass ociated with the small- est Pisot number , which is defined by U 0 = 1 , U 1 = 2 , U 2 = 3 , U 3 = 4 , U 4 = 5 , U n = U n − 1 + U n − 5 for n ≥ 5, since U n = U n − 2 + U n − 3 holds only for n ≡ 1 mod 3, n ≥ 4. For ev ery x ∈ Z ∗ , there exists y ∈ {− 1 , 0 , 1 } ∗ with x ∼ S y , k y k ≤ k x k , since 2 ∼ S 10, 20 ∼ S 1000, 200 ∼ S 1010, 20 3 ∼ S 10100, 20 4 ∼ S 100100, 20 5 ∼ S 1010 4 , 20 n ∼ S 10 6 10 n − 5 for n ≥ 6. Pr oposition 6.5. Every N ∈ Z has a unique r epresentatio n N = P n j = 1 y j S n − j with y 1 6 = 0 and y 1 · · · y n ∈ {− 1 , 0 , 1 } ∗ avoiding the set X = { 10 6 1 , 10 k 1 , 10 k ¯ 1 , 0 ≤ k ≤ 5 , and their opposites } , with the e xception that 10 6 1 , 10 5 1 , 10 5 ¯ 1 , 10 4 ¯ 1 and th eir opposites ar e possible suffixes of y 1 · · · y n . As for the Fibonacci numeration system, Proposition 6.5 is proved by considering g n , the smallest positi ve inte ger with an expan sion of length n s tarting with 1 a void ing 24 Christiane Frougny and W olfgang Steiner 0 , 0 − 1 /β 5 , 0 − 1 /β 4 , 0 − 1 /β 3 , 0 − 1 /β 2 , 0 − 1 /β , 0 − 1 , 0 − β, 0 − β 2 , 0 − 1 /β 5 , − 1 − 1 /β 4 , − 1 − 1 /β 3 , − 1 − 1 /β 2 , − 1 − 1 /β , − 1 − 1 , − 1 − β, − 1 − 1 , 1 − β, 1 − 1 /β , 1 1 /β 5 , 0 1 /β 4 , 0 1 /β 3 , 0 1 /β 2 , 0 1 /β , 0 1 , 0 β, 0 β 2 , 0 1 /β 5 , − 1 1 /β 4 , − 1 1 /β 3 , − 1 1 /β 2 , − 1 1 /β , − 1 1 , − 1 β, − 1 1 , 1 β, 1 1 /β , 1 ¯ 1 | ¯ 1 0 | 0 1 | 1 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 0 | ¯ 1 0 | 1 0 | ¯ 1 0 | 1 0 | ¯ 1 0 | 1 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 0 | ¯ 1 0 | 1 0 | 0 0 | 0 0 | 1 0 | ¯ 1 ¯ 1 | 0 1 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 0 | ¯ 1 0 | 1 1 | 0 ¯ 1 | 0 0 | 0 0 | 0 0 | ¯ 1 0 | 1 Figur e 14. Tra nsducer N β normalizing β - expan sions of minimal weight, β 3 = β + 1. these factors, and G n , the larg est integer of this kind. The representations of g n + 1 and G n , n ≥ 1, depending on the congruence class of n modulo 8 are giv en by the follo wing table. n ≡ j mod 8 g n + 1 G n g n + 1 − G n 1 , 2 , 3 , 4 1 ( 0 6 ¯ 10 ) n/ 8 ( 10 7 ) n/ 8 1 ¯ 10 j − 1 ∼ S 1 5 1 ( 0 6 ¯ 10 ) ( n − 5 ) / 8 0 4 ¯ 1 ( 10 7 ) ( n − 5 ) / 8 10 4 1 ¯ 1000 ¯ 1 ∼ S 1 6 1 ( 0 6 ¯ 10 ) ( n − 6 ) / 8 0 5 ¯ 1 ( 10 7 ) ( n − 6 ) / 8 10 5 1 ¯ 10000 ¯ 1 ∼ S 1 ¯ 1 ∼ S 1 7 1 ( 0 6 ¯ 10 ) ( n − 7 ) / 8 0 6 ¯ 1 ( 10 7 ) ( n − 7 ) / 8 10 5 1 1 ¯ 100000 ¯ 2 ∼ S 10 ¯ 2 ∼ S 1 0 1 ( 0 6 ¯ 10 ) n/ 8 ( 10 7 ) n/ 8 − 1 10 6 1 1 ¯ 100000 ¯ 1 ¯ 1 ∼ S 10 ¯ 1 ¯ 1 ∼ S 1 For the ca lculation of g n + 1 − G n we hav e used S n − S n − 1 − S n − 7 = S n − 8 for n ≥ 9. Since S n = S n − 2 − S n − 3 holds only for n ≥ 4 and not for n = 3, determining when x ∼ S y is more complicated than for ∼ F and ∼ T . If z = a 1 · · · a n . = m 3 m 2 m 1 a n . , then we have N = P n j = 1 a j S n − j = 4 m 3 + 3 m 2 + 2 m 1 + a n . W e hav e to distinguish between dif ferent v alues of a n . • If a n = 0, then N = 0 if and only if m 2 = 2 m ′ 2 , m 1 = − 2 m 3 − 3 m ′ 2 , hence z = m 3 ( β 3 − 2 β ) + m ′ 2 ( 2 β 2 − 3 β ) = − m 3 /β 4 − m ′ 2 ( 1 /β 4 + 1 /β 7 ) . In particular , m ′ 2 = 0 , m 3 ∈ { 0 , ± 1 } implies N = 0 if z ∈ { 0 , ± 1 / β 4 } . Minimal weight expa nsions in Pisot bases 25  − β 2 − 1 /β β 2 +1 , − 1 /β 2 β 2 +1   − β − 1 / β 2 β 2 +1 , − β 2 − 1 /β β 2 +1   − β 2 β 2 +1 , 1 /β 3 β 2 +1   β 2 β 2 +1 , − 1 /β 3 β 2 +1   β + 1 /β 2 β 2 +1 , β 2 +1 /β β 2 +1   β 2 +1 /β β 2 +1 , 1 /β 2 β 2 +1  0 . ¯ 1(0 5 ¯ 10 2 ) ω , . 1(0 5 10 2 ) ω . (0 ¯ 10 6 ) ω , . (10 7 ) ω . ( ¯ 10 7 ) ω , . (010 6 ) ω 0 10 5 ¯ 10 5 10 6 ¯ 10 6 0 2 0 2 010 5 , 0 ¯ 10 6 0 ¯ 10 5 , 010 6 Figur e 15. B ranching transformation and corresponding automaton, β 3 = β + 1. • If a n = 1, then N = 0 if and only if m 2 = 2 m ′ 2 − 1, m 1 = − 2 m 3 − 3 m ′ 2 + 1, z = m 3 ( β 3 − 2 β )+ m ′ 2 ( 2 β 2 − 3 β ) − β 2 + β + 1 = − m 3 /β 4 − m ′ 2 ( 1 /β 4 + 1 /β 7 )+ 1 /β 2 . In particular , m 3 m ′ 2 ∈ { 00 , ¯ 11 , 01 } provides N = 0 if z ∈ { 1 /β 2 , 1 / β 3 , 1 / β 5 } . • If a n = 2, then m 3 m 2 m 1 ∈ { 00 ¯ 1 , ¯ 101 } provides N = 0 if z ∈ { 2 − β , 1 } . W e ha ve x 1 · · · x n ∼ S y 1 · · · y n if the corresponding path in R β ends in a state z corre- sponding to a n = x n − y n (or in − z , a n = y n − x n ) and obta in the follo wing theorem. Theor em 6.6. The s et of S -e xpansions of minimal w eight in {− 1 , 0 , 1 } ∗ is r ecognized by M S , w hich is the automaton in F igure 11 including the dashed arrows. The words given by Pr oposition 6.5 ar e S -expan sions of minimal weight. For deta ils on the proof of Theorem 6.6, we refer again to [13]. 6.4 W eight of the expan sions Let W n be the set of words x = x 1 · · · x n ∈ {− 1 , 0 , 1 } n a voiding the factors giv en by Proposition 6.5. Then the sequence of random v ariables ( Y j ) 1 ≤ j ≤ n defined by Pr [ Y 1 = y 1 · · · y 7 , . . . , Y j = y j · · · y j + 6 ] = # { x 1 · · · x n + 6 ∈ W n 0 6 : x 1 · · · x j + 6 = y 1 · · · y j + 6 } / # W n 26 Christiane Frougny and W olfgang Steiner is Marko v with transition probabilities Pr [ Y j + 1 = v | Y j = u ] = p u,v + O ( β − n + j ) , ( p u,v ) u,v ∈ { 10 6 ,..., 0 6 1 , 0 7 , 0 6 ¯ 1 ,..., ¯ 10 6 } =                        0 · · · · · · 0 2 β 3 1 β 7 0 · · · 0 1 . . . . . . 0 0 . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 0 0 . . . . . . . . . 0 1 2 β 5 1 β 1 2 β 5 0 . . . . . . . . . 0 0 0 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . 0 0 . . . . . . 1 0 · · · 0 1 β 7 2 β 3 0 · · · · · · 0                        . The left eigen vector to the eigen va lue 1 of this matrix is 1 14 + 4 β 2 ( 1 , . . . , 1 , 4 β 2 , 1 , . . . , 1 ) , and we obtain the follo wing theorem (with 1 7 + 2 β 2 ≈ 0 . 09515). Theor em 6. 7. F or positive inte gers M , we have , as M → ∞ , 1 2 M + 1 M X N = − M k N k S = 1 7 + 2 β 2 log M log β + O ( 1 ) . 7 Concluding re marks Another example of a number β < 2 of small deg ree satisfying (D 2 ), which is not studied in this article, is the Pisot number satisfying β 3 = β 2 + 1, with 2 = 100 . 0000 ¯ 1. A q uestion which is not ap proached in this paper con cerns β -expansions of minima l weight in { 1 − B , . . . , B − 1 } ∗ when β does not satisfy (D B ), in particular minimal weight ex pansions on the alphabet {− 1 , 0 , 1 } when β < 3 and (D 2 ) does not hold. In view of applications to cryptography , we present a summary of the av erage min- imal weight of representations of integers in linear numeration systems ( U n ) n ≥ 0 asso- ciated with dif ferent β , with digits in A = { 0 , 1 } or in A = {− 1 , 0 , 1 } . U n A β a verage k N k U for N ∈ {− M , . . . , M } 2 n { 0 , 1 } 2 ( log 2 M ) / 2 2 n {− 1 , 0 , 1 } 2 ( log 2 M ) / 3 F n { 0 , 1 } 1 + √ 5 2 ( log β M ) / ( β 2 + 1 ) ≈ 0 . 398 log 2 M F n {− 1 , 0 , 1 } 1 + √ 5 2 ( log β M ) / 5 ≈ 0 . 288 log 2 M T n {− 1 , 0 , 1 } β 3 = β 2 + β + 1 ( log β M ) β 3 / ( β 5 + 1 ) ≈ 0 . 321 log 2 M S n {− 1 , 0 , 1 } β 3 = β + 1 ( log β M ) / ( 7 + 2 β 2 ) ≈ 0 . 235 log 2 M Minimal weight expa nsions in Pisot bases 27 If we want to compute a s calar multiple of a group element, e.g. a point P on an elliptic curve, we can choose a representation N = P n j = 0 x j U j of the scalar, compute U j P , 0 ≤ j ≤ n , by using the r ecurrence of U and finally N P = P n j = 0 x j ( U j P ) . In the cases which we ha ve considere d, this amounts to n + k N k U additions (or subtractions). Since n ≈ log β N is larger than k N k U , the smallest number of additions is usually gi ven by a 2-expansion of minimal weight. (W e hav e log ( 1 + √ 5 ) / 2 N ≈ 1 . 44 log 2 N , log β N ≈ 1 . 137 log 2 M for the T ribonacci number , log β N ≈ 2 . 465 log 2 N for the smallest Pisot number .) If ho wev er we ha ve to compute se veral multiple s N P with the same P and dif f erent N ∈ {− M , . . . , M } , then it suffices to compute U j P for 0 ≤ j ≤ n ≈ log β M once, and do k N k U additions for each N . Starting from 10 multiples of the same P , the Fibonacci numeratio n system is preferable to base 2 since ( 1 + 10 / 5 ) log ( 1 + √ 5 ) / 2 M ≈ 4 . 321 log 2 M < ( 1 + 10 / 3 ) log 2 M . Starting from 20 multiples of the same P , S - expan sions of minimal weigh t are preferable to the Fibonacci numera tion system since ( 1 + 20 / ( 7 + 2 β 2 )) log β M ≈ 7 . 156 log 2 M < 7 . 202 log 2 M ≈ ( 1 + 20 / 5 ) log ( 1 + √ 5 ) / 2 M . Refer ences [1] S. Akiyama, Cubic P isot units with finite beta expansion s , Algebraic number theory and Dio- phantine analysis (Graz, 1998 ) , de Gruyter, Berlin, 20 00, pp. 11–26. [2] S. Arno and F . S. Wheeler , Signed Digit Repr esentations of Minimal Hamming W eight , IEEE T rans. Computers 42 (1993), pp. 1007–1010 . [3] A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris Sér . A 285 (1977), pp. 419– 421. [4] A. D. Booth, A signed binary multiplication technique , Quart. J. Mech. Appl. Math. 4 (1951), pp. 236–24 0. [5] W . Bosma, Sign ed bits and fast e xponentiation , J. Thé or . Nombres Bordeaux 13 (2001), pp. 27– 41. [6] K. Dajani, C. Kraaikamp, and P . Li ardet, Erg odic pr operties of signed binary expansions , Discrete Contin. Dyn. Syst. 15 (2006), pp. 87–11 9. [7] V . Dimitrov , T . Cookle v , and B. Done vsky , Number theor etic transforms over the Golden Sec- tion quadr atic field , IEEE T rans. S ignal Proc. 43 (1995 ), pp. 1790–1797. [8] M. Drmota and W . Steiner , The Zeckendo rf expansio n o f po l ynomial sequences , J. T héor . Nom- bres Bordeaux 14 (2002), pp. 439–4 75. [9] N. Ebeid and M. A. Hasan, On τ -adic r epr esentations of inte gers , Des. Codes Cryptogr . 45 (2007), pp. 271– 296. [10] S. Eilenberg, Automata, lan guages, and machines. Vol. A . Academic Press, New Y ork, 1974. [11] Ch. Frougny , Repres entations of numbers and finite automata , Math. Systems Theory 25 (1992), pp. 37–6 0. [12] Ch. Fr ougn y and B. Solomyak, F i nite beta-expansions , E rgodic Theory Dynam. Systems 12 (1992), pp. 713– 723. [13] Ch. Frougny and W . Steiner , Minimal w eight expansions in Pisot bases , preliminary version, arXiv :0803.2874 v2. 28 Christiane Frougny and W olfgang Steiner [14] P . Grabner and C. Heuberger , On the number of optimal base 2 r epr esentations of inte gers , Des. Codes Cryptogr . 40 (2006), pp. 25–39. [15] C. Heub erger , Minimal e xpansions in r edundant number systems: Fi bonacci base s and gr eedy algorithms , Period. Math. Hungar . 49 (2004), pp. 65– 89. [16] C. Heuberg er and H. Prodinger , Analysis of alternative digit sets for nonadjacent re pr esenta- tions , Monatsh . Math. 147 (2006), pp. 219–2 48. [17] M. Joye and C. T ymen, Compact encoding of non-adjacent forms with applications to elliptic curve cryptogr aphy , Public key cryptograph y ( Cheju Island, 2001), L ecture Notes in Comput. Sci. 1992 , Springer , Berlin, 2001, pp. 353–36 4. [18] M. L othaire, Algebr aic combinatorics on wor ds , Encyclopedia of Mathematics and its Appli- cations 90. Cambridge Uni versity Press, Cambridge, 2002. [19] N. Meloni, New point addition formulae for ECC applications , Arithmetic of finite fields, Lecture Notes in Comput. Sci. 4547 , S pringer , Berlin, 2007, pp. 189–201. [20] F . Morain and J. Oliv os, Speeding up the computations on an elliptic curve using add ition- subtrac tion chains , RAIR O Inform. Théor . Appl. 24 (1990), pp. 531–543. [21] J. A. Muir and D. R. S tinson, Minimality and other pr operties of the width- w nonadjacen t form , Math. Comp. 75 (2006), pp. 369– 384. [22] W . Parry , On the β -expansions of r eal number s , Acta Math. Acad. Sci. Hungar . 11 (1960), pp. 401–41 6. [23] G. W . Reitwiesner , Binary arithmetic , Advance s in computers, Vol. 1, Academic Press, New Y ork, 1960, pp. 231–308. [24] A. Rényi, Repr esentations for r eal number s and their erg odic pr operties , Acta Math. Acad. Sci. Hungar 8 (1957), pp. 477–4 93. [25] J. Sakaro vitch, Eléments de théorie des automates . V uibert, 2003, English translation: Ele- ments of Automa t a Theory , Cambridge Univ ersity Press, to appear . [26] J. A. Solinas , Effic i ent arithmetic o n Koblitz cu rves , Des. Codes Cryptogr . 19 (2000), pp. 195 – 249. A uthor informa tion Christiane Frougn y, LIAF A , CNRS UMR 708 9, and Univ ersit é Paris 8, France. Email: chri stiane.frougn y@liafa.jussie u.fr W olfgang Steiner, LIAF A, CNRS UMR 7089, Univ ersité Paris Diderot – Paris 7, Case 70 14, 752 05 Paris Cede x 13, Fr ance. Email: stei ner@liafa.jus sieu.fr