GENERALIZED RADIX REPRESENTATIONS AND DYNAMICAL SYSTEMS IV

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1 GENERALIZED RADIX REPRESENTATIONS AND DYNAICAL SYSTES IV SHIGEKI AKIYAA, HORST BRUNOTTE, ATTILA PETHŐ, AND JÖRG THUSWALDNER Abstract For r r,, r d R d the mapping τ r : Z d Z d given by τ ra,, a d a,, a d, r a + + r d a d where denotes the floor function, is called a shift radix system if for each a Z d there exists an integer k > 0 with τr k a 0 As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β- expansions and canonical number systems After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions In particular we establish the distribution of Pisot polynomials with and without the finiteness property F Introduction This is the fourth part of a series of papers that is devoted to the systematic study of socalled shift radix systems Shift radix systems are dynamical systems that are strongly related to well-known notions of number systems First of all, let us recall their exact definition Definition Let d be an integer and r r,, r d R d To r we associate the mapping τ r : Z d Z d in the following way: For a a,, a d Z d let τ r a a,, a d, ra, where ra r a + + r d a d, ie, the inner product of the vectors r and a We call τ r a shift radix system SRS for short if for all a Z d we can find some k > 0 with τ k r a 0 In Part I [] cf also [], where some preliminary studies are contained of this series we proved that SRS form a common generalization of canonical number systems in residue class rings of polynomial rings see [9] for a definition as well as β-expansions of real numbers which were first studied in [0] and are defined below Furthermore, some partial results are given that point out the difficulty of characterizing all SRS parameters A thorough study of the SRS parameters in dimension d is done in Part II [3], while Part III [4] shows that CNS polynomials can be used in order to approximate the set of SRS parameters The present paper is devoted to the relation between β-expansions and SRS The following classes of sets are needed for our studies For d N, d let D d : { r R d : a Z d the sequence τ k r a k 0 is ultimately periodic } and D 0 d : { r R d : a Z d k > 0 : τ k r a 0 } Date: August 7, athematics Subject Classification A63, K06, R06 Key words and phrases beta expansion, canonical number system, periodic point, contracting polynomial, Pisot number The first author was supported by the Japan Society for the Promotion of Science, Grant-in Aid for fundamental research 85400, The third author was supported partially by the project JP-6/006 and by the Hungarian National Foundation for Scientific Research Grant No T67580 The fourth author was supported by project S960 of the Austrian Science Foundation The third and fourth author are supported by the Stiftung Aktion Österreich-Ungarn project number 67öu The function : R Z denotes the floor function which is defined by x : max{n Z : n x}

2 S AKIYAA, H BRUNOTTE, A PETHŐ, AND J THUSWALDNER D d is strongly related to the set of contracting polynomials In particular, let E d r : { r,, r d R d : X d + r d X d + + r has only roots y C with y < r } In [, Lemmas 4, 4 and 43] we proved that int D d E d D 0 d is the set of all parameters r Rd that give rise to an SRS As β-expansions form a central object in the investigations done in the present paper we give their exact definition cf for instance [6, 8, 0] Before that we recall the definition of Pisot and Salem numbers Let P X X d X d Z[X] be an irreducible polynomial over Z If P has a real root greater than one and all other roots are located in the open unit disk then P is called a Pisot polynomial The dominant root is called a Pisot number If P has a real root greater than one and all other roots are located in the closed unit disk and at least one of them has modulus then P is called a Salem polynomial The dominant root is called a Salem number Let β > be a non-integral real number and let A {0,,, β } be the set of digits Then each γ [0, can be represented uniquely as a β-expansion by 3 γ a m β m + a m β m + with a i A such that 4 0 γ m a i β i < β n in holds for all n m Since by condition 4 the digits a i are selected as large as possible, the representation in 3 is often called the greedy expansion of γ with respect to β Schmidt [] proved that in order to get ultimately periodic expansions for all γ Q 0, it is necessary for β to be a Pisot or a Salem number We are interested in base numbers β which give rise to finite β-expansions for large classes of numbers Let Finβ be the set of positive real numbers having finite greedy expansion with respect to β We say that β > has property F if Finβ Z[/β] [0,, that is, all reasonable candidates admit finite β-expansions It is shown in [6, Lemma ] that F can hold only for Pisot numbers β In [, Theorem ] property F is related to the SRS property We recall this in more detail Associated to Pisot and Salem numbers with periodic β-expansions and with property F, respectively, we define for each d N, d the sets B d : {,, Z d : X d X d is a Pisot or Salem polynomial} and B 0 d : {,, Z d : X d X d is a Pisot polynomial with property F} We obviously have B 0 d B d Let us consider the map ψ : B d R d defined as follows If,, B d then let β be the dominant root of the Pisot or Salem polynomial X d X d Now let ψ,, r d,, r, where r j b j β + b j+ β + + β j d j d In other words, the numbers r,, r d are defined in a way that they satisfy the relation X d X d X βx d + r X d + + r d As,, B d, the polynomial X d + r X d + + r d has all its roots in the closed unit circle Together with this implies that ψb d D d

3 RADIX REPRESENTATIONS AND DYNAICAL SYSTES 3 oreover Theorem of [] implies that ψb 0 d D 0 d We push the relation between Pisot numbers, property F and SRS further in the present paper and show that ψb d and ψbd 0 are excellent approximations of D d and Dd 0, respectively To formulate our main results we need some notation For N >0 we set 5 B d : { b,, Z d } :, b,, B d and 6 B 0 d : { b,, Z d :, b,, B 0 d} With these notations we are able to state the following theorem Theorem Let d We have 7 B d d λ d D d O /d, where λ d denotes the d -dimensional Lebesgue measure As we do not have enough information about the structure of Dd 0, we are not able to prove an asymptotic estimate for the error term for Bd 0 However, we are able to establish the main term, more precisely we prove: Theorem 3 Let d We have Bd 0 8 lim d λ d Dd 0 These results are analogous to [4, Theorems 4 and 6], where corresponding results are proved for canonical number system polynomials Notice that λ d D d is equal to, 4, 6 3, and 35 for d,, 3, 4 and 5, respectively This is trivial for d and, while the other three values were computed by Paul Surer, to whom we are much indebted for this information Properties of two auxiliary mappings In all what follows let d For Z define the mapping χ : R d Z d such that if r r d,, r R d then let χ r b,,, where, b i r i + r r i+ +, i,, d, r d + r + It is easy to check that if b,, B d, then χ b ψb b, ie, χ b is one of the left inverses of the mapping ψ We pointed out above that ψb d D d and ψbd 0 D0 d To prove the main theorem we need some properties of the sets S d : χ D d and Sd 0 : χ Dd 0 as well as S d : S d and Sd 0 : Sd 0 Z Z Our first lemma shows that if is large enough then the polynomials associated to the elements of S d behave in some sense similar as Pisot or Salem polynomials However, the example r 09999, , 9998 shows that the polynomial associated to χ r is not necessarily a Pisot or Salem polynomial if r D d Indeed, we have χ 800 r 800, 5394, 539, 797 and the polynomial X 4 800X X 539X +797 has two real roots 084, , which are larger than one Lemma There exist constants 0 > 0, c c d and c c d such that the following is true: Let Z,,, S d and P X X d X d If 0 then P X has a real root β for which the inequalities hold β < c and β b < c + O b

4 4 S AKIYAA, H BRUNOTTE, A PETHŐ, AND J THUSWALDNER Proof In this proof the constants implied by the O-notation depend only on d There exists r d,, r D d such that b,, χ r d,, r It is easy to see that r i d Thus b i r i + O, i,, d Put QX b X d + +, ie, let P X X d X d QX Then P Q and 3 P + t t + t d Q + t Assume that > 0 and Q > 0 for large enough As Q +t d d +t d we have P +t > 0 provided that t d d Thus P X has a real root in the interval, +d d Now we assume that Q < 0 for all large enough By 3 we have P + t < 0 if t d d Thus P X has again a real root, this time in the interval d d, The cases < 0 can be handled similarly Thus we proved with c d d The relation P β 0 implies Thus β + b β + b 3 β + + β d β b βb + b 3 β β + + β d Using this expression, inequality, and the estimates b i d, i,, d we get β b c d + d d c c + d b j3 c j < c + O, which proves the second assertion of the lemma Now we are in the position to extend the definition of ψ from the set B d to S d Indeed, for,, S d \ B d consider the polynomial b P X X d X d Select a real root β of P X in the following way: Then let if < 0 then choose β to be some root of P X, otherwise choose β to be a root of P X that satisfies and of Lemma ψ,, r d,, r, where the real numbers r,, r d are defined in a way that they satisfy the relation X d X d X βx d + r X d + + r d We also introduce another mapping, which yields vectors with rational coordinates approximating ψ,, good enough provided that,, S d It is easy to see that there exists a constant d > 0 such that b b holds for each where Let ψ : Sd Q d be defined by ψ,,,, S d S d : {,, S d : } + b, + b + b,, b + b + b 3 b The next lemma shows that if,, S d with large enough then ψ,, is a good approximation of ψ,, We actually prove

5 RADIX REPRESENTATIONS AND DYNAICAL SYSTES 5 Lemma Let,, S d and assume that is large enough Then ψ,, ψ,, < c 3 + O 3, where c 3 as well as the implied constant depend only on d Proof Using Lemma we estimate the distance of the coordinates starting by the first one β + b β + b β b d c < c d + O b < c 3 b + O 3 We proceed with the second coordinate and get β + β + b b < β β b + b + β β b b < < c 3 b d c d + O 3 b c + O b + d β b c β c Finally we turn to the general case In the next inequalities we have j d b j β + b j+ β + + β d+ j b j b j+ + b b b j < β c β b + b j+ β β b b + b j+ d β 3 + b k β k+ j kj+3 d c < c + O b + c d β c d β b + c + O 3 < c 33 b + O 3 Putting c 3 max{c 3, c 3, c 33, } we proved the statement In the next lemma we show that the set ψ S d is lattice-like ore precisely we prove Lemma 3 Let b,,, b b,, b d S d such that there exists a j {,, d} such that b i b i, i j and b j b j + Then 0, if j > and k d j +, d j +, ψ,, k ψb,, b + Ob, if j >, k d j + or j, k d, d k Ob, if j >, k d j + or j, k < d, k+ + O 3, if j b Here v k denotes the k-th coordinate of the vector v The implied constants depend only on d

6 6 S AKIYAA, H BRUNOTTE, A PETHŐ, AND J THUSWALDNER Proof If j > then ψb and ψb differ only in the d j + -st and d j + -nd coordinates Comparing these coordinates we obtain b j b j + b + b + b b + Ob for the d j + -st coordinate and b j b b j b b for the d j + -nd coordinate In contrast, if j {, } then all coordinates are changing Consider first the case j, ie, b b + and b j b j, j If k then we get b d b + b b b + b + b+ + b If < k < d then b k b + b b Finally, if k d then b b + b b + b k+ b b k + b If j k then we have b d b + b b b + b + b 3 b b + b + b + b + Ob b k+ b b 3 b b k + b+ b k + b Ob b + + b b + + b + b b + b + + Ob + + b + + b b + b b + + b b O 3 b The estimates for the other coordinates in the case j are obtained in the same way as in the case j 3 Proof of Theorem 3 We start with the proof of Theorem 3 An essential fact is that the region Dd 0 can be approximated by a finite union of rectangles [a, [a, b [a d,

7 RADIX REPRESENTATIONS AND DYNAICAL SYSTES 7 with arbitrarily small error from above and from below This means that Dd 0 is Jordan measurable Before we prove this fact we recall that a set X in R d is Jordan measurable if for any positive ε there exists finite set of rectangles P i i,, p and Q j j,, q satisfying Q j X P i j i and µ d i P i \ j Q j < ε Here µ d is the Jordan measure, a finitely additive measure that satisfies d µ d b k a k k Obviously Jordan measurability of X implies Lebesgue measurability and µ d X λ d X holds where λ d is the d-dimensional Lebesgue measure It is well known that X is Jordan measurable if and only if X is Jordan measurable and µ d X 0, ie, the boundary of X has measure zero It is easy to prove the following result Lemma 3 Dd 0 is Jordan measurable, ie, µ d Dd 0 0 Proof We use the same terminology as in [4]: D d,ε { r R d : ρr < ε } for ε 0, where ρr is the maximal modulus of all the roots of X d + r d X d + + r with r r, r,, r d We put Dd,ε 0 D d,ε Dd 0 Then we have essentially shown in [] that D0 d,ε is Jordan measurable In fact, D d,ε is a finite union of algebraic sets and therefore D d,ε is Jordan measurable Dd,ε 0 is also Jordan measurable because it is a subset of D d,ε that emerges from D d,ε by removing finitely many convex polyhedra Since E d is Jordan measurable, for a given positive ε, there is a finite union of rectangles such that E d int with µ d < ε/ There exists a positive κ such that Now implies E d + κ \ E d κ D 0 d D 0 d, κ D 0 d \ D 0 d, κ D 0 d D 0 d, κ Therefore there is a finite union of rectangles such that D 0 d with µ d < ε Now we are in a position to prove Theorem 3 For some rectangle [u, v [u 3, v 3 [u d, v d in R d let A : {b,, Z d To prove 8 it is enough to show that lim : ψ, b,, } A d d v i u i, when E 3 Indeed, by the above Lemma 3 one can approximate D0 d by a finite disjoint union of rectangles E 3 from below and from above with an arbitrarily small error We note that ψ : B d R d is obviously injective Thus when we fix > 0, each lattice point b,, with, b,, d d B d is in one to one correspondence with a point ψ, b,, which is close to b,, Therefore instead of counting the points of the shape ψ, b,, contained in we count the number of points in Zd Lemma implies that r j bj + O, j,, d, we see that this causes an error O d and we get { A b,, Z d b :,, b d } + O d i

8 8 S AKIYAA, H BRUNOTTE, A PETHŐ, AND J THUSWALDNER Indeed, the term O d estimates the number of lattice points of the form b,, which have distances O from Here the implied constants depend only on d Combining this with the trivial observation { lim d b,, Z d b :,, b d d } v i u i, we get the assertion 4 Auxiliary lemmata In order to show Theorem we need two preliminary lemmata which are stated and proved in the present section We start with a lemma that quantifies the continuous dependence of the roots of a polynomial from its coefficients Lemma 4 Let d N and ρ, ε R >0 Then there exists a constant c 4 > 0 depending only on d and ρ with the following property: if all roots α C of the polynomial P X X d + p d X d + + p 0 R[X] satisfy α < ρ and QX X d + q d X d + + q 0 R[X] is chosen such that p i q i < ε, i 0,, d then for each root β of QX there exists a root α of P X satisfying 4 β α < c 4 ε /d In particular, all roots β of QX satisfy β < ρ + c 4 ε /d Proof Let α,, α d denote the roots of P X and fix an arbitrary root β of QX Let k {,, d} Then d p d k α i α ik ρ k k This implies i < <i k d d q d k ρ k + ε k By a well-known theorem of Cauchy [7, Corollary 54] we get { β max d q d k /k} c 5 ρ k d with a constant c 5 depending only on d Choose the root α of P X such that β α is minimal among the differences β α j Then on one hand d d Qβ P β ε β j ε c 5 ρ j c 6 ε On the other hand j0 Qβ P β j0 d β α j β α d j Comparing the last two inequalities we get 4 with c 4 c /d 6 for the chosen β But since β was an arbitrary root of QX this proves the result The next lemma contains a refinement of Lemma 47 of [4] for D d Lemma 4 Let 0 < η < Then we have λ d E d + η \ D d dd+/ λ d E d η i and λ d D d \ E d η dd+/ λ d E d η

9 RADIX REPRESENTATIONS AND DYNAICAL SYSTES 9 Proof First we express E d t with the help of E d for any positive real number t Indeed let r,, r d E This means that the roots of the polynomial P X X d + r d X d + + r lie in the open unit circle Then the roots of P X t td X d + r d t X d + + r are of absolute t d value at most t, ie, the point r t d,, r d t belongs to E d t Obviously this mapping is bijective, which means we have E d t diagt d,, te d, where diagv,, v d denotes the d-dimensional diagonal matrix with entries v,, v d This implies 4 λ d E d t t dd+/ λ d E d We prove only the first relation, because the second one can be done similarly Let 0 < η < Setting t + η in 4 we get immediately λ d E d + η \ D d λ d E d + η \ E d λ d E d + η λ d E d + η dd+/ λ d E d dd+/ λ d E d η This proves the first assertion The second one follows similarly 5 Proof of Theorem It is possible to prove Theorem without error term following the line of the Section 3 However, we are able to give a bound for the error term in this case This makes the proof of Theorem much more involved Before starting with this proof we introduce some notation Let > 0 and put and Then we claim W x, s {y R d : x y s/} x R d, s R W d x B d 5 λ d W d B d d W ψx, + O Indeed, let be large enough and x, y B d such that x y e j for some j {,, d} Then by Lemmata and 3 Thus ψx k ψy k ψx k ψx k + ψx k ψy k + ψy k ψy k { + O, if j, k d j + O, otherwise 5 λ d W ψx, W ψy, O d As x has at most d neighbors we get W ψx, W ψy, Bd O λ d x,y B d x y and the claim is proved Now we are in the position to give a lower bound for λ d D d Let x B d such that ψx E d c4 /d D d Let y W ψx, Then ρψx < c 4 /d and as ψx y we get ρy < by Lemma 4 Thus 53 W ψx, D d x B d ρψx< c 4 /d d

10 0 S AKIYAA, H BRUNOTTE, A PETHŐ, AND J THUSWALDNER Putting η c 4 /d, Lemma 4 implies that the measure of the set D d \ E d c 4 /d is bounded by O /d oreover this set satisfies the conditions of the Theorem of H Davenport [5] Observe that h and V m of [5] are in the actual application independent from Thus the number of x B d such that c 4 /d ρψx is at most O d /d Combining this with 5 and 53 we obtain the desired lower bound 54 λ d D d B d d c 7 /d Here, c 7 > 0 is a constant To prove an upper bound we need some preparation, more precisely we will construct for every r r d,, r D d and a large enough integer vector b,, Z d such that ψb is located near enough to r Indeed put b χ r and consider ψb + b, + b + b,, b + b + b 3 b We estimate the distance of the coordinates of this vector to r Putting k r r 3 k r + for d we have k O and As r + k < b b r + k + for d > and + r r d < + r r d + we obtain b { } d + b r d max + r r d + / + r b + k / r d, + r r d / + r + k + / r d This implies after a short computation r d + b + O A similar calculation proves for j,, d This means b j + b + b j+ b r j + O ψb r + O Applying now Lemma we obtain ψb r ψb r + ψb ψb + O Thus by Lemma 4 note that ψb, r E d we get ρψb ρr + c 4 /d + c 4 /d

11 RADIX REPRESENTATIONS AND DYNAICAL SYSTES for large enough This means that if is large enough then all but one root of X d X d have absolute value at most + c 4 /d and one root is close to We have further W ψx, D d x Z d ψx E d +c 4 /d x B d W ψx, x Z d ψx E d +c 4 /d \E d W ψx, This time we apply Lemma 4 with η c 4 /d and conclude that the volume of the set E d + c 4 /d \ D d is at most O /d As the conditions of the Theorem of Davenport [5] hold again we get that the number of x Z d such that ψx lies in E d + c4 /d \ D d is at most O d /d Thus there is a constant c 8 > 0 such that λ d D d B d d + c 8 /d Comparing this inequality with 54 we obtain Theorem References [] S Akiyama, T Borbély, H Brunotte, A Pethő and J Thuswaldner, On a generalization of the radix representation a survey, in High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst Commun, 4 004, 9 7 [] S Akiyama, T Borbély, H Brunotte, A Pethő and J Thuswaldner, Generalized radix representations and dynamical systems I, Acta ath Hungar, , [3] S Akiyama, H Brunotte, A Pethő and J Thuswaldner, Generalized radix representations and dynamical systems II, Acta Arith 006, 6 [4] S Akiyama, H Brunotte, A Pethő and J Thuswaldner, Generalized radix representations and dynamical systems III, Osaka J ath , [5] H Davenport, On a principle of Lipschitz J London ath Soc 6, Corrigendum ibid , 580 [6] C Frougny and B Solomyak, Finite beta-expansions, Ergod Th and Dynam Sys 99, [7] ignotte and D Ştefănescu, Polynomials: An Algorithmic Approach, Springer, 999 [8] W Parry, On the β-expansions of real numbers, Acta ath Acad Sci Hungar 960, [9] A Pethő, On a polynomial transformation and its application to the construction of a public key cryptosystem, Computational Number Theory, Proc, Eds: A Pethő, Pohst, H G Zimmer and H C Williams, Walter de Gruyter Publ Comp 99, 3 43 [0] A Rényi, Representations for real numbers and their ergodic properties, Acta ath Acad Sci Hungar, 8 957, [] K Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull London ath Soc, 980, [] I Schur, Über Potenzreihen, die im Inneren des Einheitskreises beschränkt sind II, J reine angew ath, 48 98, 45 Department of athematics, Faculty of Science Niigata University, Ikarashi -8050, Niigata 950-8, JAPAN address: akiyama@mathscniigata-uacjp Haus-Endt-Strasse 88, D Düsseldorf, GERANY address: brunoth@webde Department of Computer Science, University of Debrecen, Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen PO Box, H-400 Debrecen, HUNGARY address: pethoe@infunidebhu Chair of athematics and Statistics, University of Leoben, Franz-Josef-Strasse 8, A-8700 Leoben, AUSTRIA address: JoergThuswaldner@mu-leobenat