9 0 0 2 n Minimal weight expansions in Pisot bases a J 9 ChristianeFrougny and WolfgangSteiner ] M D Abstract.Forapplicationstocryptography,itisimportanttorepresentnumberswithasmallnumber ofnon-zerodigits(Hammingweight)orwithsmallabsolutesumofdigits. Theproblemoffinding . s representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent c forminbase2.Inthispaper,weconsidernumerationsystemswithrespecttorealbasesβwhichare [ Pisotnumbersandprovethattheexpansionswithminimalabsolutesumofdigitsarerecognizable byfiniteautomata.WhenβistheGoldenRatio,theTribonaccinumberorthesmallestPisotnumber, 3 wedetermineexpansionswithminimalnumberofdigits±1andgiveexplicitelythefiniteautomata v recognizingalltheseexpansions.Theaverageweightislowerthanforthenon-adjacentform. 4 7 Keywords.Minimalweight,beta-expansions,Pisotnumbers,Fibonaccinumbers,automata. 8 2 AMSclassification.11A63,11B39,68Q45,94A60. . 3 0 8 1 Introduction 0 : Let A be a set of (integer) digits and x = x x x be a word with letters x in A. v Theweightofxistheabsolutesumofdigits1 x2··=· n n x . TheHammingjweight Xi of x is the number of non-zero digits in x. Okf ckoursPe, jw=h1e|njA| 1,0,1 , the two ⊆ {− } r definitionscoincide. a Expansions of minimal weight in integer bases β have been studied extensively. When β = 2, it is known since Booth [4] and Reitwiesner [23] how to obtain such anexpansionwiththedigitset 1,0,1 . Thewell-knownnon-adjacent form(NAF) {− } isaparticular expansion ofminimal weight withthepropertythatthe non-zerodigits are isolated. It has many applications to cryptography, see in particular [20, 17, 21]. Other expansions of minimal weight in integer base are studied in [14, 16]. Ergodic propertiesofsignedbinaryexpansionsareestablishedin[6]. Non-standard number systems — where the base is not an integer — have been studied from various points of view. Expansions in a real non-integral base β > 1 have been introduced by Rényi [24] and studied initially by Parry [22]. Number the- oretic transforms where numbers are represented in base the Golden Ratio have been introduced in [7] for application to signalprocessing and fast convolution. Fibonacci representations have been used in [19] to design exponentiation algorithms based on addition chains. Recently, the investigation of minimal weight expansions has been extended totheFibonacci numeration systembyHeuberger [15], whogavean equiv- alenttotheNAF. Solinas [26]hasshownhowtorepresentascalar inacomplex base τ relatedtoKoblitzcurves,andhasgivenaτ-NAFform,andtheHammingweightof theserepresentationshasbeenstudiedin[9]. ThisresearchwassupportedbytheFrenchAgenceNationaledelaRecherche, grantANR-JCJC06-134288 “DyCoNum”. 2 ChristianeFrougny and WolfgangSteiner In this paper, we study expansions in a real base β > 1 which is not an integer. Any number z in the interval [0,1) has a so-called greedy β-expansion given by the β-transformation τ , which relies on a greedy algorithm: let τ (z) = βz βz and β β −⌊ ⌋ define, for j ≥ 1, xj = ⌊βτβj−1(z)⌋. Then z = ∞j=1xjβ−j, where the xj’s are integer digits in the alphabet 0,1,..., β . We wrPite z = .x1x2 . If there exists { ⌊ ⌋} ··· a n such that x = 0 for all j > n, the expansion is said to be finite and we write j z =.x x x . Byshifting,anynon-negativerealnumberhasagreedyβ-expansion: 1 2 n ··· Ifz [0,βk),k 0,andβ kz =.x x ,thenz =x x .x x . − 1 2 1 k k+1 k+2 ∈ ≥ ··· ··· ··· Weconsiderthesequences ofdigitsx x aswords. Sincewewanttominimize 1 2 ··· the weight, we are only interested in finite words x = x x x , but we allow a 1 2 n ··· priori arbitrary digits x in Z. The corresponding set of numbers z = .x x x is j 1 2 n therefore Z[β 1]. Note that we do not require that the greedy β-expansion o··f·every − z Z[β 1] [0,1) is finite, although this property (F) holds for the three numbers β − ∈ ∩ studiedinSections4to6,see[12,1]. ThesetoffinitewordswithlettersinanalphabetAisdenotedbyA ,asusual. We ∗ definearelationonwordsx=x x x Z ,y =y y y Z by 1 2 n ∗ 1 2 m ∗ ··· ∈ ··· ∈ x y ifandonlyif .x x x =βk .y y y forsomek Z. β 1 2 n 1 2 m ∼ ··· × ··· ∈ A word x Z is said to be β-heavy if there exists y Z such that x y and ∗ ∗ β ∈ ∈ ∼ y < x . Wesaythatyisβ-lighterthanx. Thismeansthatanappropriateshiftofy k k k k providesaβ-expansionofthenumber.x x x withsmallerabsolutesumofdigits 1 2 n ··· than x . If x is not β-heavy, then we call x a β-expansion of minimal weight. It is k k easytoseethateverywordcontaining aβ-heavyfactor isβ-heavy. Thereforewecan restrictourattentiontostrictlyβ-heavywordsx=x x Z ,whichmeansthatx 1 n ∗ ··· ∈ isβ-heavy,andx x andx x arenotβ-heavy. 1 n 1 2 n ··· − ··· Inthefollowing,weconsiderrealbasesβ satisfyingthecondition (D ): thereexistsawordb 1 B,...,B 1 suchthatB band b B B ∗ β ∈{ − − } ∼ k k≤ for some positive integer B. Corollary 3.2 and Remark 3.4 show that every class of words (with respect to ) contains a β-expansion of minimal weight with digits in β ∼ 1 B,...,B 1 ifandonlyifβ satisfies(D ). B { − − } Ifβ isaPisotnumber,i.e.,analgebraicintegergreaterthan1suchthatalltheother rootsofitsminimalpolynomialareinmoduluslessthanone,thenitsatisfies(D )for B someB >0byProposition3.5. Thecontraryisnottrue: Thereexistalgebraicintegers β > 1 satsfying (D ) which are not Pisot, e.g. the positive solution of β4 = 2β +1 B isnotaPisotnumber butsatisfies(D )since2 = 1000.( 1). Thefollowingexample 2 − providesalargeclassofnumbersβ satisfying(D ). B Example1.1. If1 = .t t t (t )ω withintegerst t t > t 0, 1 2 d d+1 1 2 d d+1 ··· ≥ ≥ ··· ≥ ≥ thenβ satisfies(D )withB =t +1= β +1,since B 1 ⌊ ⌋ t βd+1−t1βd−···−tdβ−td+1 = βd+11 =βd−t1βd−1−···−td − andthus βd+1 (1+t )βd+(t t )βd 1+ +(t t )β+(t t )=0. 1 1 2 − d 1 d d d+1 − − ··· − − − MinimalweightexpansionsinPisotbases 3 Recall that the set of greedy β-expansions is recognizable by a finite automaton whenβisaPisotnumber[3]. Inthiswork,weprovethatthesetofallβ-expansionsof minimalweightisrecognizedbyafiniteautomatonwhenβ isaPisotnumber. We then consider particular Pisot numbers satisfying (D ) which have been exten- 2 sivelystudiedfromvariouspointsofview. Whenβ istheGoldenRatio,weconstruct afinite transducer which gives, for a strictlyβ-heavy word as input, aβ-lighter word asoutput. SimilarlytotheNon-AdjacentForminbase2,wedefineaparticularunique expansion of minimal weight avoiding a certain given set of factors. We show that there is a finite transducer which converts all words of minimal weight into these expansions avoiding these factors. From these transducers, we derive the minimal automaton recognizing the set of β-expansions of minimal weight in 1,0,1 . We ∗ {− } giveabranchingtransformationwhichprovidesallβ-expansionsofminimalweightin 1,0,1 of a given z Z[β 1]. Similar results areobtained for the representation ∗ − {− } ∈ ofintegersintheFibonacci numerationsystem. Theaverage weightofexpansions of thenumbers M,...,M is 1log M,whichmeansthattypicallyonlyeveryfifthdigit − 5 β isnon-zero. Notethatthecorrespondingvaluefor2-expansions ofminimalweightis 1log M,see[2,5],andthat 1log M 0.288log M. 3 2 5 β ≈ 2 We obtain similar results for the case where β is the so-called Tribonacci number, which satisfies β3 = β2 + β + 1 (β 1.839), and the corresponding representa- ≈ tions forintegers. Inthis case, theaverage weight is β3 log M 0.282log M β5+1 β ≈ β ≈ 0.321log M. 2 FinallyweconsiderthesmallestPisotnumber,β3 =β+1(β 1.325),whichpro- ≈ videsrepresentationsofintegerswithevenlowerweightthantheFibonaccinumeration system: 1 log M 0.095log M 0.234log M. 7+2β2 β ≈ β ≈ 2 SincetheprooftechniquesfortheTribonaccinumberandthesmallestPisotnumber are quite similar to the Golden Ratio case (but more complicated), some parts of the proofs are not contained in the final version of this paper. The interested reader can findthemin[13]. 2 Preliminaries A finite sequence of elements of a set A is called a word, and the set of words on A is the free monoid A . The set A is called alphabet. The set of infinite sequences ∗ or infinite words on A is denoted by AN. Let v be a word of A , denote by vn the ∗ concatenationofvtoitselfntimes,andbyvω theinfiniteconcatenationvvv . ··· Afinitewordvisafactorofa(finiteorinfinite)wordxifthereexistsuandwsuch that x = uvw. When u is the empty word, v is a prefix of x. The prefix v is strict if v =x. Whenwisempty,vissaidtobeasuffixofx. 6 Werecallsomedefinitionsonautomata,see[10]and[25]forinstance. Anautoma- tonoverA, =(Q,A,E,I,T),isadirectedgraphlabelledbyelementsofA. Theset A ofvertices,traditionallycalledstates,isdenotedbyQ,I Qisthesetofinitialstates, ⊂ T Q is the set of terminal states and E Q A Q is the set of labelled edges. If(⊂p,a,q) E,wewritep a q. Theautom⊂aton×isfin×iteifQisfinite. AsubsetH of ∈ → A issaidtoberecognizablebyafiniteautomatonifthereexistsafiniteautomaton ∗ A 4 ChristianeFrougny and WolfgangSteiner suchthatH is equaltothesetof labels of paths startinginaninitial stateandending inaterminalstate. Atransducerisanautomaton =(Q,A A ,E,I,T)wheretheedgesofE are ∗ ′∗ T × labelled by couples of words in A A . It is said to be finite if the set Q of states ∗ ′∗ × uv andthesetE ofedgesarefinite. If(p,(u,v),q) E, wewritep | q. Inthispaper ∈ −→ we consider letter-to-letter transducers, where the edges are labelled by elements of A A. Theinputautomatonofsuchatransducerisobtainedbytakingtheprojection ′ × ofedgesonthefirstcomponent. 3 General case Inthissection,ouraimistoprovethatonecanconstructafiniteautomatonrecognizing thesetofβ-expansionsofminimalweightwhenβ isaPisotnumber. Weneedfirstsomecombinatorialresultsforbasesβ satisfying(D ). Notethatβ is B notassumedtobeaPisotnumberhere. Proposition 3.1. If β satisfies (D ) with some integer B 2, then for every word B ≥ x Z thereexistssomey 1 B,...,B 1 withx y and y x . ∗ ∗ β ∈ ∈{ − − } ∼ k k≤k k Corollary3.2. Ifβsatisfies(D )withsomeintegerB 2,thenforeverywordx Z B ∗ ≥ ∈ thereexistsaβ-expansionofminimalweighty 1 B,...,B 1 withx y. ∗ β ∈{ − − } ∼ Remark3.3. Ifβ satisfies(D )forsomepositiveintegerB,thenitiseasytoseethat B β satisfies(D )foreveryintegerC >B. C Remark3.4. Ifβ doesnotsatisfy(D ),thenallwordsx 1 B,...,B 1 with B ∗ ∈{ − − } x Bareβ-heavierthanB. Itfollowsthatthesetofβ-expansionsofminimalweight β ∼ x Bis0 B0 . β ∗ ∗ ∼ ProofofProposition3.1. Let A = 1 B,...,B 1 . If x = x x x A , 1 2 n ∗ { − − } ··· ∈ then there is nothing to do. Otherwise, we use (D ): there exists some word b = B b b A such that b b (b B)b b 0 and b B. We use k d ∗ k 1 0 1 d β th−is·r·e·latio∈n to decrease the−ab·so··lut−e valu−e of a d·i·g·it x∼ A withkoukt≤increasing the h 6∈ weightofx,andweshowthatweeventuallyobtainawordinA ifwealwayschoose ∗ the rightmost such digit. More precisely, set x(0) = x for 1 j n, x(0) = 0 for j j ≤ ≤ j j 0 and j > n, b = 0 for j < k and j > d. Define, recursively for i 0, j ≤ − ≥ h =max j Z: x(i) B , i { ∈ | j |≥ } x(i+1) =x(i)+sgn(x(i))(b B), x(i+1) =x(i) +sgn(x(i))b forj =0, hi hi hi 0− hi+j hi+j hi j 6 aslongash exists. Thenwehave x(0) = x , x(i+1)β j = x(i)β j and i jP∈Z| j | k k jP∈Z j − jP∈Z j − x(i+1) = x(i+1) + x(i+1) x(i) + b B+ (x(i) + b ) x(i) . | j | | hi | | hi+j|≤| hi| | 0|− | hi+j| | j| ≤ | j | Xj Z Xj=0 Xj=0 Xj Z ∈ 6 6 ∈ MinimalweightexpansionsinPisotbases 5 wIfihthioduotetshenoletaedxiinsgt,atnhdentrwaielinhgavzeer|xos(ji)i|si ,m 0,thus m m≥0 im ≤ ℓ m ≥ x him−1 x(im+1) him−1−1 x(im−1+1) + k b (m+1) k b . k k≥ j=X | j |≥ j=X | j | Xj=1| −j|≥···≥ Xj=1| −j| −∞ −∞ If k b > 0, this is not possible since x is finite. Similarly, (h ) has no j=1| −j| k k i i≥0 maPximum if d b > 0. Since x(i+1) can differ from x(i) only for h k j j=1| j| j j i − ≤ ≤ hi+d,wehaPvehi+1 hi+dforalli 0. Ifhi < hi′,i < i′,thenthereistherefore ≤ ≥ a sequence (i ) , i i < i < < i = i, with M (h h )/d m 0 m M 0 1 M ′ i′ i such that h h≤ f≤or all ℓ≤ i ,i +1·,·.·..,i , m 0,...,M ≥. As ab−ove, we im ≤ ℓ ∈ { m m ′} ∈ { } obtain x (M+1) k b ,butM canbearbitrarilylargeif(h ) hasneither minimukmkn≥or maximumP.j=H1e|n−cej|we have shown that hi cannot existifio≥r0all i 0 if k b >0and d b >0. ≥ PjI=t1re|m−aji|ns to consPidje=r1th|ej|case b = B with k = 0 or d = 0. Assume, w.l.o.g., k k d= 0. Thenwehaveh h . Ifh existsforalli 0,thenboth k x(i) and ∞j=1|x(hii)+j|areeventuia+l1ly≤conistant.iThereforewem≥usthavesomePi,ij′=w0i|thhhi−i′j|k·k·.·xT(hhii)i,sixm(hiip′′)l+i1exs(hxii′(h′)i+i)−2k······=x(hi0i)h∼i−βh0i′xor(hii)β+h1ix−(hhii)i+′2=··1·., In the first case, x(i) x(i) without the trailing zeros is a word y A with the hi+1 hi+2··· ∈ ∗ desiredproperties. Inthelattercase,eachx Z canbeeasilytransformedintosome ∗ ∈ y 1,0,1 withy xand y = x ,andthepropositionisproved. 2 ∗ β ∈{− } ∼ k k k k Thefollowingpropositionshowsslightlymorethantheexistenceofapositiveinte- gerB suchthatβ satisfies(D )whenβ isaPisotnumber. B Proposition 3.5. For every Pisotnumber β, there exists some positive integer B and somewordb Z suchthatB band b w ,thenwehave y y > x x +w . Sinces =(x y ) (x y )., j s 1 j 1 j s j 1 1 j j k ··· k k ··· k − ··· − 8 ChristianeFrougny and WolfgangSteiner the digitwise subtraction of 0max(i j,0)x x 0m i and 0max(j i,0)a a provides − 1 j − − 1 m ··· ··· a word which is β-lighter than y y , which contradicts the assumption that y is a 1 j ··· β-expansionofminimalweight. Let W = max w sisastatein . If δ W B, then let h j be such s β j { | A } ≤ − − ≤ that x = 0, x = 0 for h < i j. Since x < B, we have δ < δ +B h i h h 1 j W 6 w , hence x x ≤ > y | y| +w . Let−a a be th≤e w−ord≤wh−ichshw−a1s used fokr1th·e··dehfi−n1iktion kof1w··· h,−i1.ke., s sh−=1 a 1a··.a· m a , w = a a . Thenthedigitwiseadditsiho−n1of0max(hi−h1+1,0)y1···yi i+01m··i·anmd 0msahx−(1h 1 ki,01)a··· mak providesawordwhichisβ-lighterth−anx 1x··· h.−S1ince−x = − − 1 m 1 h 1 h 0,thiscontradic·t·s·theassumptionthatxisstrictlyβ-heavy. ··· − 6 Let betherestrictionof tothestates(s,δ)with W B <δ w withsome β β s S T − − ≤ additionalinitialandterminalstates: Everystatewhichcanbereachedfrom(0,0)by a path with input in 0 is initial, and every 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 consists only of β-heavy words and contains all strictly β-heavy words in A . β ∗ S ThereforeM =A A HA isthesetofβ-expansionsofminimalweightinA ,and ∗ ∗ ∗ ∗ \ M isrecognizablebyafiniteautomatonbyLemma3.9. 2 Theorem 3.11. Let β be a Pisot number. Then one can construct a finite automaton recognizingthesetofβ-expansionsofminimalweight. Proof. Proposition 3.5 shows that β satisfies (D ) for some positive integer B, and B that no β-expansion of minimal weight y Z can contain a digit y with y B, ∗ j j ∈ | | ≥ sinceweobtainaβ-lighterwordifwereplaceBbybasintheproofofProposition3.1. ThereforeTheorem3.10impliesTheorem3.11. 2 4 GoldenRatiocase Inthissectionwegiveexplicitconstructionsforthecasewhereβ istheGoldenRatio 1+√5. Wehave1 = .11, hence 2 = 10.01andβ satisfies(D ), seealsoExample1.1. 2 2 Corollary 3.2 shows that every z Z[β 1] can be represented by a β-expansion of − ∈ minimal weight in 1,0,1 . For most applications, only these expansions are in- ∗ {− } teresting. Remark that the digits of arbitrary β-expansions of minimal weight are in 2, 1,0,1,2 bytheproofofTheorem3.11,since3=100.01. {−For−typograp}hicalreasons,wewritethedigit 1as1¯ inwordsandtransitions. − 4.1 β-expansionsofminimalweightforβ = 1+√5 2 Our aim in this section is to construct explicitly the finite automaton recognizing the β-expansionsofminimalweightinA ,A= 1,0,1 . ∗ {− } Theorem4.1. Ifβ = 1+√5,thenthesetofβ-expansionsofminimalweightin 1,0,1 2 {− }∗ isrecognizedbythefiniteautomaton ofFigure1whereallstatesareterminal. β M MinimalweightexpansionsinPisotbases 9 1 100 1 0 1 0 0¯10 ¯10 10 010 0 ¯1 1 0 0 0 0 ¯1 0 0 ¯1 0100 00 0 00 0¯100 ¯1 ¯100 Figure1. Automaton recognizingβ-expansionsofminimalweightforβ = 1+√5 Mβ 2 (left)andacompactrepresentationof (right). β M ItisofcoursepossibletofollowtheproofofTheorem3.10,butthestatesof are β A 1 1 1 1 1 1 1 0, , , , 1, β, β2, β , β , β2 , β2 , ±β3 ±β2 ±β ± ± ± ± ± β2 ± ± β3 ± ± β2 ± ± β3 thus W = 2 and the transducer has 160 states. For other bases β, the number of β S statescanbemuchlarger. Thereforewehavetorefinethetechniquesifwedonotwant computer-assisted proofs. Itis possibleto show that a large partof is not needed, β S e.g. by excluding some β-heavy factors such as 11 from the output, and to obtain finally the transducer in Figure 2. However, it is easier to prove Theorem 4.1 by an indirectstrategy,whichincludessomeresultswhichareinterestingbythemselves. Lemma4.2. Allwordsin 1,0,1 whicharenotrecognizedbytheautomaton ∗ β {− } M inFigure1areβ-heavy. Proof. Thetransducer inFigure2isapartofthetransducer intheproofof Theo- β S rem3.10. Thismeansthateverywordwhichistheinputofapath(withfullordashed transitions) going from (0,0) to (0, 1) is β-heavy, because the output has the same − value but less weight. Since a β-heavy word remains β-heavy if we omit the leading andtrailingzeros,thedashedtransitionscanbeomitted. Thenthesetofinputsis H =1(0100) 1 1(0100) 0101 1(001¯0) 1¯ 1(001¯0) 01¯ ∗ ∗ ∗ ∗ ∪ ∪ ∪ 1¯(01¯00) 1¯ 1¯(01¯00) 01¯01¯ 1¯(0010) 1 1¯(0010) 01 ∗ ∗ ∗ ∗ ∪ ∪ ∪ ∪ and isconstructedasintheproofofLemma3.9. 2 β M SimilarlytotheNAFinbase2, where theexpansions ofminimal weightavoid the set 11,11¯,1¯1¯,1¯1 ,weshowinthenextresultthat, forβ = 1+√5,everyrealnumber { } 2 admitsaβ-expansionwhichavoidsacertainfinitesetX. Proposition 4.3. If β = 1+√5, then every z R has a β-expansion of the form 2 ∈ z = y y .y y with y 1,0,1 such that y y avoids the set 1 k k+1 k+2 j 1 2 X = 11··,·101,1001,11¯,·1··01¯, andthe∈iro{p−posites}. If z Z[β] =··Z·[β 1], then this − { } ∈ expansionisuniqueuptoleadingzeros. 10 ChristianeFrougny and WolfgangSteiner 0|1 0|¯1 −1,1 0,0 1,1 1|0 ¯1|0 −1,0 −1/β,−1 1/β,−1 1,0 1|0 0|0 0|¯1 0|0 ¯1|0 1|0 0|0 0|1 0|0 ¯1|0 −1/β,0 −1,−1 1,−1 1/β,0 1|0 1|0 ¯1|0 ¯1|0 0|¯1 0|1 0,−1 −1/β,−2 1/β,−2 0,−1 Figure2. Transducerwithstrictlyβ-heavywordsasinputs,β = 1+√5. 2 Proof. We determine this β-expansion similarly to the greedy β-expansion in the In- troduction. Notethatthemaximalvalueof.x x forasequencex x avoiding 1 2 1 2 theelementsofX is.(1000)ω =β2/(β2+1). Ifw··e·definethetransforma·t·io·n β2 β2 β2 β2 β2+1 τ : − , − , , τ(z)=βz z+1/2 , (cid:20)β2+1 β2+1(cid:19)→(cid:20)β2+1 β2+1(cid:19) −(cid:22) 2β (cid:23) aynd=se1tyfojr=so(cid:4)mβe22+βj1τj−11,(tzh)e+n w1/e2h(cid:5)afvoerτzj∈(zh)β−2β+β21,ββ2+21β(cid:17),j, ≥β21,then1z==.y11y2,·1·/·β. If, j ≥ ∈ × β2+1 β2+1 − β−2+1 β2+1 hence yj+1 = 0, yj+2 = 0, and τj+2(z) ∈ β−2β+21,β(cid:2)2β+1 , hence(cid:1)yj+3 ∈(cid:2){1¯,0}. Th(cid:1)is showsthatthegivenfactors areavoided. As(cid:2)imilarargu(cid:1)ment fory = 1showsthat j − theoppositesareavoidedaswell,hencewehaveshowntheexistenceoftheexpansion for z ∈ hβ−2β+21,ββ2+21(cid:17). For arbitrary z ∈ R, the expansion is given by shifting the expansionofβ kz,k 0,totheleft. − Ifwechoosey =0≥incaseτj 1(z)>β/(β2+1)=.(0100)ω,thenitisimpossible j − to avoid the factors 11, 101 and 1001 in the following. If we choose y = 1 in case j τj 1(z) < β/(β2+1),thenβτj 1(z) 1 < 1/(β2+1) = .(001¯0)ω,andthusitis − − impossibletoavoid thefactors 11¯, 101¯−, 1¯1¯, 1¯0−1¯ and 1¯001¯. Since β/(β2+1) Z[β], wehaveτj 1(z)= β/(β2+1)forz Z[β]. Similarrelationsholdfortheopp6∈osites, − 6 ∈ thustheexpansionisunique. 2 Remark 4.4. Similarly, the transformation τ(z) = βz z +1/2 on [ β/2,β/2) providesforeveryz Z[β]auniqueexpansionavoiding−th⌊efactors⌋11,10−1,11¯,101¯, 1001¯ andtheiroppos∈ites. Proposition 4.5. If x is accepted by , then there exists y 1,0,1 avoiding β ∗ X = 11,101,1001,11¯,101¯ andtheirMopposites with x y∈a{n−d x =} y . The β { } ∼ k k k k transducer inFigure3realizestheconversionfrom0x0toy. β N

Minimal weight expansions in Pisot bases

Similar Minimal weight expansions in Pisot bases