Acta Polytechnica DOI:10.14311/AP.2020.60.0214 Acta Polytechnica 60(3):214–224, 2020 © Czech Technical University in Prague, 2020 available online at https://ojs.cvut.cz/ojs/index.php/ap BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES Jonathan Caalima, Shiela Demegilloa, b, ∗ a University of the Philippines Diliman, Institute of Mathematics, C.P. Garcia, 1101 Quezon City, Philippines b Adamson University, Mathematics and Physics Department, San Marcelino St., 1000 Manila, Philippines ∗ corresponding author: ssdemegillo@upd.edu.ph Abstract. We introduce a numeration system, called the beta Cantor series expansion, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix γ ∈ R and a sequence B of real number bases, every element of the interval [γ,γ + 1) has a beta Cantor series expansion with respect to B where the digits are integers in some alphabet A(B). We give a criterion in determining whether an integer sequence is admissible when B satisfies some condition. We provide a description of the reference strings, namely the expansion of γ and γ + 1, used in the admissibility criterion. Keywords: Beta expansion, Q-Cantor series expansion, numeration system, admissibility. 1. Introduction The subject of representations of real numbers is an extensively studied research field. In the seminal work [1], Renyi introduced the now well-known con- cept of beta expansions. Beta expansions are repre- sentations of real numbers using an arbitrary positive real base β > 1 obtained via the beta transformation Tβ : [0, 1) −→ [0, 1) given by Tβ(x) = βx−bβxc. The iterates of T induce a numeration system on [0, 1) wherein the expansion of an element x ∈ [0, 1) is given by the sequence d(β; x) = (d1,d2, . . . ) with di = bβTi−1(x)c. Thus, the digits di belong to the alphabet A = {0, 1, . . . ,bβc} if β /∈ N or A = {0, 1, . . . ,β − 1} if β ∈ N. Parry, in [2], considered the admissibility problem of determining the integer sequences over the alphabet A that appear as the beta expansion of a real number in the domain [0, 1). Parry provided a necessary and sufficient condition (formulated in terms of the beta expansion of 1) for a sequence of integers to be beta admissible. In the subsequent paper [3], Parry extended the definition of the beta transformation to T : [0, 1) −→ [0, 1) where T(x) = βx + α + bβx + αc with β > 1 and 0 ≤ α < 1 and he also tackled the admissibility problem in this setting. An important generalization of beta expansion is a positional numeration system that uses negative bases. As remarked by Frougny and Lai in [4], it appears that Grünwald was the first to introduce this idea in [5]. Here, we present a general formulation considered by Ito and Sadahiro in [6]. Let 1 < β ∈ R and define lβ := −β/(β + 1) and rβ := 1/(β + 1). The negative beta transformation is the map T−β : [lβ,rβ) −→ [lβ,rβ) given by T−β(x) = −βx−b−βx− lβc. The map T−β also induces an expansion on the domain [lβ,rβ), where the digits are given by b−βTi(x) − lβc. An admissibility criterion was also given in [6, Theorem 10]. (In [7], Liao and Steiner introduced the self-map T̂ : (0, 1] −→ (0, 1] given by T̂(x) = −βx + bβxc + 1. This transformation is conjugate to the one defined by Ito and Sadahiro and, as such, the results for the negative beta expansion can be restated using the map T̂.) As with the positive beta transformations, Dombek, et.al, in [8] introduced a parameter α to generalize the negative beta transformation defined by Ito and Sadahiro. They considered the map T : [α,α + 1) −→ [α,α + 1) given by T(x) = −βx−b−βx−αc where β > 1 and α ∈ (−1, 0]. (See also [9, 10] for other transformations inducing an expansion in a negative base.) The motivation of the current study originates from a certain class of rotational beta expansions in dimen- sion two (see [11, 12]). Rotational beta expansions generalize the notion of beta expansions in higher dimensions. Let Z = [0, 1) × [0, 1) and 1 < β ∈ R. Define the four-fold rotational beta transformation T : Z −→Z by T ([ x y ]) = [ −βy −b−βyc βx−bβxc ] . It is easy to see that if we wish to keep track of the itinerary of a point z ∈ Z under T, then we need to alternatingly apply the functions f1(x) = −βx−b−βxc and f2(x) = βx−bβxc to an element x ∈ [0, 1). This series of applications of the maps f1 and f2 yields a numeration system in [0, 1) in two bases −β and β as discussed in Section 2 below. This numeration system is akin to the Q-Cantor series expansion [13]. Given a sequence Q = (qn)n≥1 of integers qn ≥ 2, the Q-Cantor expansion of a real 214 https://doi.org/10.14311/AP.2020.60.0214 https://ojs.cvut.cz/ojs/index.php/ap vol. 60 no. 3/2020 Beta Cantor Series Expansion and Admissible Sequences number x is the unique expansion of the form x = E0 + ∑ n≥1 En Πnj=1qj where E0 = bxc and En ∈ {0, 1, . . . ,qn − 1} for all n ≥ 1 such that En 6= qn − 1 infinitely many number of times. We call the numeration system considered in this paper the beta Cantor series expansion as it marries the notions of beta expansion and Q-Cantor series expansion. As mentioned in Section 2, the beta ex- pansion of Parry and the negative beta expansion of Ito and Sadahiro are examples of beta Cantor series expansion. It is the hope of the authors that the beta Cantor series expansion provides a unified formulation for the positive and negative beta numeration systems to further highlight their similarities. After all, the positive and negative beta expansions share many similar properties (see e.g. [9, 14, 15]). Our goal is to extend the work of Parry on admissibility to beta Cantor series expansions. In Section 2, we define the transformations that induce the beta Cantor series expansion. In Section 3, we provide a discussion on the relationship between two different definitions of the expansion of γ + 1 (similar to the expansion of 1 in [2] and the expansion of rβ in [6]). In Section 4, we tackle the problem of finding a necessary and sufficient condition for a sequence to be admissible with respect to the beta Cantor series expansion. 2. B-expansion maps Fix γ ∈ R and let B = (β1,β2, . . . ) where βi ∈ R for all i ∈ N. For j ∈ N, we define fj : [γ,γ + 1) −→ [γ,γ + 1) by fj(x) = βjx −bβjx−γc. For m ∈ N, consider the transformation Tm = TmB = T m B,γ on [γ,γ + 1) given by Tm(x) = fm(. . .f3(f2(f1(x))) . . . ). Hence, Tm(x) = βmTm−1(x) −am(x) where am(x) = ⌊ βmT m−1(x) −γ ⌋ . For β = βm, we also define uβ := min{bβγ −γc ,bβ(γ + 1) −γc}, vβ := max{bβγ −γc ,bβ(γ + 1) −γc} and A(β) := { [uβ,vβ) ∩Z if β > 0 and β + γ(β − 1) ∈ Z [uβ,vβ] ∩Z otherwise. Then am(x) ∈A(βm). Define A(B) := Π∞m=1A(βm), which is the set of all sequences (d1,d2, . . . ) where dm ∈A(βm). For ease of notation, we define B[i,j] := Πjm=iβm. When i = 1, we write B[j] instead of B[1,j] with the convention that B[0] := 1. Observe that B[m + i] = B[m]B[m + 1,m + i]. The transformations Tm induce a numeration sys- tem on the interval [γ,γ + 1) over the alphabet A(B) if limm→∞ |B[m]| = ∞. Proposition 2.1. Let B = (β1,β2, . . . ) ∈ RN and x ∈ [γ,γ + 1). If limm→∞ |B[m]| = ∞, then x = ∞∑ i=1 ai(x) B[i] . Proof. For simplicity, let aj = aj(x). Note that Tj−1(x) = Tj(x) + aj βj . Hence, x = a1 β1 + T(x) β1 = a1 β1 + a2 β1β2 + T 2(x) β1β2 . In general, x = m∑ i=1 ai B[i] + Tm(x) B[m] . This implies that as m →∞,∣∣∣∣∣x− m∑ i=1 ai B[i] ∣∣∣∣∣ = ∣∣∣∣Tm(x)B[m] ∣∣∣∣ ≤ max{|γ| , |γ + 1|}|B[m]| → 0. We write x = ∞∑ i=1 ai B[i] as (a1,a2, . . . )B. We call the sequence d(B; x) := (a1,a2, . . . ) the B-expansion of x. Let 1 < β ∈ R. Note that if γ = 0 and B = (β), then the B-expansion of x coincides with the classical β-expansion. (Here, v stands for the periodic repetition of a word v.) When γ = −β/(β + 1) and B = (−β), then the B-expansion coincides with the (−β)-expansion. If B is periodic, say B = (β1,β2, . . . ,βN ) for some N ∈ N, we also call the B- expansion as the {β1, . . . ,βN}-expansion of x. In this case, we may write d(B; x) as d(β1, . . . ,βn; x). We may extend the definition of Tj to γ + 1 as has been done in [2] and [6]. For all j ∈ N, define Tj(γ + 1) := βjTj−1(γ + 1) − ⌊ βjT j−1(γ + 1) −γ ⌋ . As in Proposition 2.1, we have γ + 1 = ∞∑ i=1 ci B[i] where ci := ⌊ βiT i−1(γ + 1) −γ ⌋ . We also write d(B; γ+1) = (c1,c2, . . . ). Note that c1 ∈ [uβ1,vβ1 ]∩Z and for j > 1, cj ∈A(βj) since Tj−1(γ + 1) < γ + 1. Example. Let α = −β = (1 + √ 5)/2 be the golden mean. Table 1 gives some information on the {α,β}- transformations for various values of γ. 215 Jonathan Caalim, Shiela Demegillo Acta Polytechnica γ A(α) A(β) d(α,β; γ) d(α,β; γ + 1) 0 {0, 1} {−2,−1, 0} (0) (1,−1, 0) 1/α {0, 1} {−4,−3,−2} (0,−3, 1,−3, 1,−2) (2,−2, 1) 2 {1, 2} {−7,−6} (1,−6, 1,−7) (2,−7, 1) Table 1. The expansion of γ and γ + 1 under various values of γ when α = −β = (1 + √ 5)/2 Figure 1. The maps T and T 2 when γ = 0, α = −β = (1 + √ 5)/2 Let us consider the particular case of γ = 0. Then fα(x) = { αx if x ∈ [0, 1/α) αx− 1 if x ∈ [1/α, 1) and fβ(x) =   −αx if x = 0 −αx + 1 if x ∈ (0, 1/α] −αx + 2 if x ∈ (1/α, 1) . Figure 1 depicts the shape of the {α,β}- transformations T and T 2. From these, we obtain a graph G (Figure 2) describing the dynamics of the map Tm. In this graph, the vertices are subintervals of [γ,γ + 1) that form its partition and there is a directed edge (dashed, if τ = α; and solid, otherwise) from vertex V1 to vertex V2 labelled d if and only if V2 ⊂ fτ (V1) and the corresponding digit is d. Now, let γ = 1/α. We have fα(x) = { αx if x ∈ [1/α, 1) αx− 1 if x ∈ [1,α) and fβ(x) =   −αx + 2 if x ∈ [1/α, 3/α− 1] −αx + 3 if x ∈ (3/α− 1, 4/α− 1] −αx + 4 if x ∈ (4/α− 1,α) . Figure 3 gives a graph corresponding to γ = 1/α where J := (1/α, 4 − 2α), K := (4 − 2α, 3/α − 1), L := (3/α−1, 1), M := (1, 2α−2), P := (2α−2, 3−α), Q := (3 −α, 4/α− 1) and R := (4/α− 1,α). 3. Expansion of γ + 1 The expansion of γ + 1 defined in the previous section proves to be insufficient for our purposes and hence, the definition needs to be modified. In this section, we present another definition of the expansion of γ + 1 analogous to those defined in [2] and [6, Lemma 6]. Hereafter, we assume B = (β1,β2, . . . ) ∈ RN with limm→∞ |B[m]| = ∞. Definition 1. We define d∗(B; γ + 1) = (c∗1,c∗2, . . . ) as the limit lim x→(γ+1)− d(B; x). That is, for any n ∈ N, there exists an �n > 0 such that for all x ∈ (γ + 1 − �n,γ + 1) and for all i < n, the i-th digit of d(B; x) is c∗i where �n+1 < �n and �n → 0 as n →∞. Example. Let β be a quadratic Pisot number. Then β satisfies the minimal polynomial x2 − bx− c where b ∈ N and 1 ≤ c ≤ b; or b ∈ N−{1, 2} and 2−b ≤ c ≤ −1. Let γ = 0. We compute for d∗(β,−β; γ + 1) = d∗(β,−β; 1). Let � > 0 be arbitrarily small. Case 1. Let 1 ≤ c ≤ b. Then b < β < b + 1. We have T (1 − �) = β (1 − �) −bβ (1 − �)c = β − �− b T 2 (1 − �) = −β (β − b− �) −b−β (β − b− �)c = � T 3 (1 − �) = β�−bβ�c = � T 4 (1 − �) = −β�−b−β�c = −� + 1. Hence, d∗(β,−β; γ + 1) = (b,−c, 0,−1). It can be shown that d(β,−β; γ + 1) = (b,−c, 0). 216 vol. 60 no. 3/2020 Beta Cantor Series Expansion and Admissible Sequences (0, 1 α2 ) ( 1 α2 , 1 α ) ( 1 α , 1) 0 1α2 1 α 0 0 0 1 1 −1 −1 −1 −2 −2 −1 1 0 0 0 −1 Figure 2. Tα,β : [0, 1) −→ [0, 1) with α = −β = (1 + √ 5)/2 J K L M P Q R 0 0 0 01 1 1 1 1 1 1 −2 −2 −2 − 3 −3 −3−3 −3 −3 −3 −4 −4 Figure 3. Tα,β : [1/α,α) −→ [1/α,α) with α = −β = (1 + √ 5)/2 Case 2. Let 2−b ≤ c ≤−1. Then b−1 < β < b. So T (1 − �) = β (1 − �) −bβ (1 − �)c = β − �− b + 1 T 2 (1 − �) = −β (β − b + 1 − �) −b−β (β − b + 1 − �)c = −β + b + � T 3 (1 − �) = β(−β + b + �) −bβ(−β + b + �)c = � T 4 (1 − �) = −β�−b−β�c = −� + 1. Hence, d∗(β,−β; γ + 1) = (b− 1,−b− c,−c,−1). Also, we have d(β,−β; γ + 1) = (b− 1,−b− c,−c, 0). Example. Let B = (α,β) where α ∈ Z < 0 and 0 > β ∈ R. Suppose α(γ + 1) − γ ∈ Z and T 2n(γ + 1) = fn(β) − bfn(β) −γc and T 2n+1(γ + 1) = gn(β) − bgn(β) −γc for some polynomials fn and gn of degree n in Z[x] where fn(β) − γ /∈ Z and gn(β) − γ /∈ Z for all n ∈ N (e.g. β may be taken to be transcendental over Q and γ ∈ Q). Let d(α,β; γ + 1) = (c1,c2, . . . ) and d∗(α,β; γ + 1) = (c∗1,c∗2, . . . ). Then, for small � > 0, c∗1 = bα(γ + 1) −γ + �c = α(γ + 1) −γ = c1. Since fn(β)−γ and gn(β)−γ are not integers, we can show that Tn(γ + 1−�) = Tn(γ + 1) + (−1)n+1�. Moreover, c∗2n = bfn(β) −γ − �c = bfn(β) −γc = c2n and c∗2n+1 = bgn(β) −γ + �c = bgn(β) −γc = c2n+1. Hence, d(α,β; γ + 1) = d∗(α,β; γ + 1). From the examples above, we see that d(B; γ + 1) may or may not be equal to d∗(B; γ + 1). In what follows, we characterize the B-expansions such that d(B; γ + 1) = d∗(B; γ + 1). Let sgn denote the signum function. Define IB := {n ∈ N∪{0} | sgn(B[n + 1]) > 0} and CB := {γ ∈ R | βn+1Tn(γ + 1) −γ /∈ Z for n ∈ IB}. Proposition 3.1. If γ ∈ CB, then d(B; γ + 1) = d∗(B; γ + 1). Proof. We show by induction on n ∈ N∪{0} that, for arbitrarily small constant, � > 0, Tn(γ + 1 − �) = Tn(γ + 1) −B[n]�. (?) The case where n = 0 is clear. Suppose (?) holds for some n ∈ N∪{0}. Then Tn+1(γ + 1 − �) = βn+1Tn(γ + 1 − �) −bβn+1Tn(γ + 1 − �) −γc = βn+1Tn(γ + 1) −B[n + 1]� −bβn+1Tn(γ + 1) −B[n + 1]�−γc . Since γ ∈ CB and � is arbitrarily small, then bβn+1Tn(γ + 1) −B[n + 1]�−γc = bβn+1Tn(γ + 1) −γc. Therefore, Tn+1(γ + 1 − �) = βn+1Tn(γ + 1) −B[n + 1]� −bβn+1Tn(γ + 1) −γc = Tn+1(γ + 1) −B[n + 1]�. Thus, we get d(B; γ + 1) = d∗(B; γ + 1). Proposition 3.2. If d(B; γ + 1) = d∗(B; γ + 1), then γ ∈CB. Proof. Suppose d(B; γ + 1) = (c1,c2, . . . ) and d∗(B; γ + 1) = (c∗1,c∗2, . . . ) are equal. We show, by induction on n ∈ N∪{0}, that Tn(γ + 1 − �) = Tn(γ + 1) −B[n]�. (?) 217 Jonathan Caalim, Shiela Demegillo Acta Polytechnica The base case n = 0 is clear. Suppose (?) holds for some n ∈ N∪{0}. Then Tn+1(γ + 1 − �) = βn+1(Tn(γ + 1) −B[n]�) − c∗n+1 = βn+1Tn(γ + 1) −B[n + 1]�− cn+1 = Tn+1(γ + 1) −B[n + 1]�. Thus, for all n ∈ N∪{0} and � > 0 sufficiently small, c∗n+1 = bβn+1T n(γ + 1) −B[n + 1]�−γc = bβn+1Tn(γ + 1) −γc = cn+1. If n ∈ IB, then βn+1Tn(γ + 1) − γ /∈ Z. Thus, γ ∈ CB. Combining Prop. 3.1 and Prop. 3.2, we have the following theorem. Theorem 3.3. γ ∈ CB if and only if d(B; γ + 1) = d∗(B; γ + 1). Theorem 3.3 and [2, Theorem 3] imply Corollary 3.3.1 while Theorem 3.3 and [6, Lemma 6] imply Corol- lary 3.3.2. Corollary 3.3.1. Let 1 < β ∈ R. Let T : [0, 1) −→ [0, 1) be the beta transformation given by T(x) = βx−bβxc. Then the following are equivalent: (1.) d(B; 1) = d∗(B; 1); (2.) βTj(1) /∈ Z for all j ∈ N∪{0}; (3.) d(B; 1) is infinite. Corollary 3.3.2. Let 1 < β ∈ R. Let T−β be the negative beta transformation on [lβ,rβ) given by T−β(x) = −βx−b−βx− lβc. Then the following are equivalent: (1.) d(B; rβ) = d∗(B; rβ); (2.) −βT 2j+1−β (rβ) − lβ /∈ Z for all j ∈ N∪{0}; (3.) d(B; rβ) is not purely periodic of odd period. Next, we determine the relation between d∗(B; γ+1) and d(B; γ + 1) when they are not equal (i.e., γ /∈CB). Define the propositional statement E(B; k) to mean βk+1T k(γ + 1) −γ ∈ Z and sgn(B[k + 1]) > 0. Suppose E(B; k) holds and k is minimal with such property. Then Tk+1(γ+1) = βk+1Tk(γ+1)−bβk+1Tk(γ+1)−γc = γ. Thus, if d(B; γ + 1) = (c1,c2, . . . ), then d(B; γ + 1) = (c1,c2, . . . ,ck+1) ◦d(σk+1(B); γ), where ◦ denotes the usual word concatenation and σj (j ∈ N) is the shift operator in RN given by σj(r1,r2, . . . ) = (rj+1,rj+2, . . . ). Moreover, from the proof of Proposition 3.1, we see that Tk+1(γ + 1 − �) = βk+1Tk(γ + 1) −B[k + 1]� −bβk+1Tk(γ + 1) −B[k + 1]�−γc = γ + 1 − �. Therefore, d∗(B; γ+1) = (c1,c2, . . . ,ck+1−1)◦d∗(σk+1(B); γ+1). From the computation above, we see that the process of determining d∗(B; γ + 1) depends on the other sequences d∗(σi(B); γ + 1), i ∈ N. To illustrate this process, we present the two-base expansion case where we set α := β1 > 0 and β := β2 > 0. We easily compute IB to be N∪{0}. Suppose E(B; k) is satisfied and k is minimal. On the one hand, suppose k is odd. Then d(α,β; γ + 1) = (c1,c2, . . . ,ck+1) ◦d(α,β; γ) and d∗(α,β; γ + 1) = (c1,c2, . . . ,ck+1−1)◦d∗(α,β; γ + 1). This implies that d∗(α,β; γ + 1) = (c1,c2, . . . ,ck+1 − 1). On the other hand, suppose k is even. Then d(α,β; γ + 1) = (c1,c2, . . . ,ck+1) ◦d(β,α; γ) and d∗(α,β; γ + 1) = (c1,c2, . . . ,ck+1−1)◦d∗(β,α; γ + 1). Note that Iσ(B) = N∪{0}. Suppose that there is no m ∈ N such that E(σ(B); m) holds. Then d∗(β,α; γ + 1) = d(β,α; γ + 1) and so, d∗(α,β; γ + 1) = (c1,c2, . . . ,ck+1 −1)◦d(β,α; γ + 1). Let d(β,α; γ + 1) = (q1,q2, . . . ). If there exists m ∈ N∪{0} such that E(σ(B); m) holds and m is minimal and odd, then d∗(β,α; γ + 1) = (q1,q2, . . . ,qm+1 − 1). Therefore, d∗(α,β; γ + 1) = (c1,c2, . . . ,ck+1 − 1) ◦ (q1,q2, . . . ,qm+1 − 1). Finally, if m is even, we have d∗(β,α; γ+1) = (q1,q2, . . . ,qm+1−1)◦d∗(α,β; γ+1). 218 vol. 60 no. 3/2020 Beta Cantor Series Expansion and Admissible Sequences Hence, d∗(α,β; γ + 1) = (c1,c2, . . . ,ck+1 − 1) ◦ (q1,q2, . . . ,qm+1 − 1) ◦d∗(α,β; γ + 1) = (c1,c2, . . . ,ck+1 − 1,q1,q2, . . . ,qm+1 − 1). To sum up, we have the following proposition. Proposition 3.4. Let B = (α,β) where α,β ∈ R > 0 and αβ > 1. Let d(α,β; γ + 1) = (c1,c2, . . . ) and d(β,α; γ + 1) = (q1,q2, . . . ). Then d∗(α,β; γ + 1) can only assume one of the following forms: (1.) (c1,c2, . . . ,c2k − 1) (2.) (c1,c2, . . . ,c2k+1 − 1,q1,q2, . . . ) (3.) (c1,c2, . . . ,c2k+1 − 1,q1,q2, . . . ,q2m − 1) (4.) (c1,c2, . . . ,c2k+1 − 1,q1,q2, . . . ,q2m+1 − 1) (5.) (c1,c2, . . . ) Examples. We now give examples to illustrate Prop. 3.4 (1–5) by providing values of α > 0 and β > 0 with αβ > 1 and γ = 0 for each case. Let r,s ∈ N. (1.) Let α = r/s /∈ Z and β = s. d(α,β; γ + 1) = (br/sc,r −sbr/sc, 0) d∗(α,β; γ + 1) = (br/sc,r −sbr/sc− 1) (2.) Let α = r and β be transcendental over Q. d(α,β; γ + 1) = (r, 0) d(β,α; γ + 1) = d∗(β,α; γ + 1) = (q1,q2, . . . ) d∗(α,β; γ + 1) = (r − 1,q1,q2, . . . ) (3.) Let α = (1 + √ 5)/2 and β = α2. d(α,β; γ + 1) = (1, 1, 1, 0) d(β,α; γ + 1) = (2, 1, 0) d∗(α,β; γ + 1) = (1, 1, 0, 2, 0) (4.) Let β = α + 1 where α is the (smallest) Pisot number which satisfies α3 −α− 1 = 0. d(α,β; γ + 1) = (1, 0, 1, 0) d(β,α; γ + 1) = (2, 0, 1, 0) d∗(α,β; γ + 1) = (1, 0, 0, 2, 0, 0) (5.) Let α be transcendental over Q and β = r. Then d(α,β; γ + 1) = d∗(α,β; γ + 1). To end this section, we recover the classical results for beta and negative beta expansions. Let 1 < β ∈ R. For the positive beta expansion, we see that IB = N∪ {0} and CB = {γ ∈ R | βTn(γ + 1)−γ /∈ Z for all n ∈ IB}. Suppose 0 /∈ CB. Then there exists a minimal k ∈ N∪{0} such that E(B; k) holds. We have d(B; 1) = (c1, . . . ,ck+1) ◦d(B; 0) = (c1, . . . ,ck+1, 0) and d∗(B; 1) = (c1, . . . ,ck+1 − 1). In other words, d∗(B; 1) ={ d(B; 1) if d(B; 1) is infinite (c1,c2, . . . ,cn − 1) if d(B; 1) = (c1, . . . ,cn, 0). For the negative beta expansion, we have IB = 2N − 1 and CB = {γ ∈ R | −βT 2n−1(γ + 1) − γ /∈ Z for all n ∈ N}. Suppose lβ /∈ CB. Then there ex- ists minimal k ∈ N such that E(B; 2k − 1) holds. Let d(B; lβ) = (a1,a2, . . . ). It is easy to see that d(B; rβ) = (0,a1,a2, . . . ). From E(B; 2k − 1), it fol- lows that d(B; rβ) = (c1, . . . ,c2k,a1,a2, . . . ). This means that c1 = 0; ci = ai−1 for all i = 2, . . . , 2k; and ai = a2k−1+i for all i ∈ N. Therefore, d(B; lβ) = (a1, . . . ,a2k−1) and d∗(B; rβ) = (c1, . . . ,c2k − 1) ◦d∗(B; rβ) = (c1, . . . ,c2k − 1) = (0,a1, . . . ,a2k−1 − 1). This is equivalent to d∗(B; rβ) =   (0,a1,a2, . . . ,a2n−1 − 1) if d(B; lβ) = (a1,a2, . . . ,a2n−1) d(B; rβ) otherwise. 4. Admissible Sequences Throughout this section, we let B = (β1,β2, . . . ) ∈ RN with limm→∞ |B[m]| = ∞. A B-representation of a real number x ∈ [γ,γ + 1) is an expansion of the form x = ∞∑ i=1 di B[i] with (d1,d2, . . . ) ∈ A(B). Note that the condition lim |B[m]| = ∞ does not guarantee that any sequence (d1,d2, . . . ) in A(B) is a B-representation of a real number x since the series ∑∞ i=1 di/B[i] may not con- verge. If the sum converges, we adopt the notation (d1,d2, . . . )B = ∑∞ i=1 di/B[i]. Now, the B-expansion of x is a particular B- representation of x. Deciding whether a sequence (d1,d2, . . . ) in A(B) is the B-expansion of an element of [γ,γ + 1), thus, entails showing that the series converges. Definition 2. An integer sequence (d1,d2, . . . ) ∈ A(B) is B-admissible if there is an x ∈ [γ,γ + 1) such that d(B; x) = (d1,d2, . . . ). 219 Jonathan Caalim, Shiela Demegillo Acta Polytechnica The admissibility of sequences with respect to the B-expansion map is related to the admissibility of se- quences for a special class of rotational beta expansion map. Let Z = [0, 1) × [0, 1) and 1 < β ∈ R. Define the map T : Z −→Z by T ((x,y)) = (−βy −b−βyc, βx−bβxc). Let T be the B-expansion map on [0, 1) with B = (−β,β). It follows that for all n ∈ N, we have T 2n−1(x,y) = ( T 2n−1B (y), T 2n−1 σ(B) (x) ) and T 2n(x,y) = ( T 2nσ(B)(x), T 2n B (y) ) . So, if d(B; y) = (a1,a2, . . . ) and d(σ(B); x) = (b1,b2, . . . ), then the expansion of (x,y) with respect to T is ((a1,b1), (b2,a2), (a3,b3), . . . ) . Proposition 4.1. Let B = (−β,β) with β > 1. Then (a1,a2, . . . ) ∈ A(B) is B-admissible and (b1,b2, . . . ) ∈A(σ(B)) is σ(B)-admissible if and only if ((a1,b1), (b2,a2), (a3,b3), . . . ) is admissible with re- spect to T . In this section, our goal is to provide an admissibility criterion for sequences in A(B). We first mention few results. Lemma 4.2. Let x ∈ [γ,γ + 1) such that d(B; x) = (a1,a2, . . . ). For n ∈ N, Tn(x) = B[n]x− n∑ i=1 aiB[n] B[i] . Proof. We prove this lemma by induction. Let x ∈ [γ,γ + 1). Then T (x) = B[1]x−a1. Suppose that for some k ∈ N, Tk(x) = B[k]x− k∑ i=1 aiB[k]/B[i]. Thus, Tk+1(x) = βk+1Tk(x) −ak+1 = B[k + 1]x− k∑ i=1 aiB[k + 1] B[i] −ak+1 B[k + 1] B[k + 1] = B[k + 1]x− k+1∑ i=1 aiB[k + 1] B[i] . In the following lemma, we give certain con- ditions for a B-representation (d1,d2, . . . ) to be a B-expansion. Note that the convergence of the sum (d1,d2, . . . )B implies the convergence of (dk+1,dk+2, . . . )σk (B) for all k ∈ N∪{0}. Lemma 4.3. Let (d1,d2, . . . ) be a B-representation of x ∈ [γ,γ + 1). If (dk+1,dk+2, . . . )σk (B) ∈ [γ,γ + 1) for all k ∈ N∪{0}, then d(B; x) = (d1,d2, . . . ). Proof. By induction on n ∈ N, we prove that dn =⌊ βnT n−1(x) −γ ⌋ and Tn(x) = (dn+1,dn+2, . . . )σn(B). Note that β1x − d1 = (d2,d3, . . . )σ(B) ∈ [γ,γ + 1). Hence, d1 = ⌊ β1T 0(x) −γ ⌋ and T(x) = (d2,d3, . . . )σ(B). Suppose the claim holds for n ≤ k where k ∈ N. Then βk+1T k(x) −dk+1 = βk+1 ( dk+1 βk+1 + dk+2 βk+1βk+2 + . . . ) −dk+1 = dk+2 βk+2 + dk+3 βk+2βk+3 + . . . = (dk+2,dk+3, . . . )σk+1(B) ∈ [γ,γ + 1). Hence, dk+1 = ⌊ βk+1T k+1(x) −γ ⌋ and Tk+1(x) = (dk+2,dk+3, . . . )σk+1(B). Corollary 4.3.1. Let x ∈ [γ,γ + 1) such that d(B; x) = (a1,a2, . . . ). Then d(σn(B); Tn(x)) = (an+1,an+2, . . . ). Proof. By Lemma 4.2, Tn(x) = B[n] ( x− n∑ i=1 ai B[i] ) = B[n] ∑ i≥n+1 ai B[i] = B[n] ∑ i≥1 an+i B[n + i] = ∑ i≥1 an+i B[n + 1,n + i] . Thus, (an+1,an+2, . . . ) is a σn(B)-representation of Tn(x). For all k ∈ N, we have σk(an+1,an+2, . . . )σk (σn(B)) = σn+k(a1,a2, . . . )σn+k (B) = Tn+k(x) ∈ [γ,γ + 1). The conclusion then follows from Lemma 4.3. Remark. Proposition 2.1, Lemma 4.2, and Corollary 4.3.1 also hold when x = γ + 1. From Lemma 4.3 and Corollary 4.3.1, we obtain the following proposition, which gives an admissibility criterion for a sequence (d1,d2, . . . ) ∈A(B) in terms of σk(d1,d2, . . . )σk (B). Proposition 4.4. A sequence (d1,d2, . . . ) ∈ A(B) is B-admissible if and only if σk(d1,d2, . . . )σk (B) ∈ [γ,γ + 1) for all k ∈ N∪{0}. Now, we provide another admissibility criterion – this time, in terms of the shifts of a sequence (x1,x2, . . . ) ∈ A(B). To this end, we need to in- troduce an order ≺B on A(B). 220 vol. 60 no. 3/2020 Beta Cantor Series Expansion and Admissible Sequences Definition 3. Let (a1,a2, . . . ) and (b1,b2, . . . ) be in A(B). We say (a1,a2, . . . ) ≺B (b1,b2, . . . ) if and only if there exists k ∈ N such that bi = ai for all i = 1, 2, . . . ,k − 1 and bk 6= ak where (bk −ak) sgn(B[k]) ≥ 1. If (a1,a2, . . . ) ≺B (b1,b2, . . . ) or (a1,a2, . . . ) = (b1,b2, . . . ), we write (a1,a2, . . . ) �B (b1,b2, . . . ). Remark. For the classical positive and negative beta expansions, the order ≺B coincides with the orders defined in [2] and [6], respectively. The following proposition tells us that the mono- tonicity of points in [γ,γ + 1) is carried over to the ordering of words with respect to ≺B. Proposition 4.5. Let x,y ∈ [γ,γ + 1). Then d(B; x) ≺B d(B; y) if and only if x < y. Proof. Let d(B; x) = (x1,x2, . . . ) and d(B; y) = (y1,y2, . . . ). Let k ∈ N be the least integer such that xk 6= yk. Suppose d(B; x) ≺B d(B; y). Then y −x = yk −xk B[k] + ∑ i≥k+1 yi −xi B[i] . We have∑ i≥k+1 yi −xi B[i] = ∑ i≥1 yk+i −xk+i B[k + i] = 1 B[k] ∑ i≥1 yk+i −xk+i B[k + 1,k + i] = Tk(y) −Tk(x) B[k] = (Tk(y) −Tk(x)) sgn(B[k]) |B[k]| > −1 |B[k]| . Thus, y −x > (yk −xk) sgn(B[k]) − 1 |B[k]| ≥ 0. For the reverse implication, suppose 0 < y −x = yk −xk + Tk(y) −Tk(x) B[k] . Note that −1 < Tk(y) − Tk(x) < 1. When sgn(B[k]) > 0, then yk − xk + 1 > 0. This implies that yk − xk ≥ 0 since both yk and xk are integers. But since yk 6= xk, then yk −xk ≥ 1. However, when sgn(B[k]) < 0, then 0 > yk−xk−1. Thus, yk−xk ≤ 0. But since yk 6= xk, then yk −xk ≤−1. In both cases, (yk −xk) sgn(B[k]) ≥ 1. Proposition 4.5, together with Corollary 4.3.1, im- plies the following result. Corollary 4.5.1. If (d1,d2, . . . ) ∈ A(B) is B- admissible, then, for all n ∈ N, d(σn(B); γ) �σn(B) (dn+1,dn+2, . . . ). Analogous to Proposition 2.1 and Lemma 4.3, we provide a relation between d∗(B; γ + 1) and γ + 1. Proposition 4.6. If d∗(B; γ + 1) = (c∗1,c∗2, . . . ), then γ + 1 = (c∗1,c∗2, . . . )B and (c∗k+1,c ∗ k+2, . . . )σk (B) ∈ [γ,γ + 1] for all k ∈ N. Proof. Suppose d∗(B; γ + 1) = (c∗1,c∗2, . . . ). Then there exist a sequence {�n} converging to 0 and a sequence {Yn} such that Yn ∈ (γ + 1 − �n,γ + 1) and d(B; Yn) = (c∗1, . . . ,c∗n,yn,1,yn,2, . . . ). Thus, Yn = n∑ i=1 c∗i B[i] + ∑ i≥1 yn,i B[n + i] . Since∑ i≥1 yn,i B[n + i] = 1 B[n] ∑ i≥1 yn,i σn(B)[i] ∈ 1 B[n] [γ,γ + 1), then lim n→∞ ∑ i≥1 yn,i B[n + i] = 0. Hence, γ + 1 = lim n→∞ Yn = lim n→∞   n∑ i=1 c∗i B[i] + ∑ i≥1 yn,i B[i]   = ∑ i≥1 c∗i B[i] . Now, for j,k ∈ N, let us consider d(B; Yk+j) = (c∗1,c ∗ 2, . . . ,c ∗ k+j,yk+j,1,yk+j,2, . . . ). By Corollary 4.3.1, d(σk(B); Tk(Yk+j)) = (c∗k+1, . . . ,c ∗ k+j,yk+j,1,yk+j,2, . . . ). Hence, (c∗k+1, . . . ,c ∗ k+j,yk+j,1,yk+j,2, . . . )σk (B) ∈ [γ,γ + 1). That is, γ ≤ wj := j∑ i=1 c∗k+i σk(B)[i] + ∑ i≥1 yk+j,i σk(B)[j + i] < γ + 1. Since {wj} tends to (c∗k+1,c ∗ k+2, . . . )σk (B), then γ ≤ (c∗k+1,c ∗ k+2, . . . ) ≤ γ + 1. Proposition 4.7. If x ∈ [γ,γ + 1), then d(B; x) ≺B d∗(B; γ + 1). 221 Jonathan Caalim, Shiela Demegillo Acta Polytechnica Proof. Let d∗(B; γ + 1) = (c∗1,c∗2, . . . ). Then there exist a sequence �n tending to zero and Yn ∈ (γ + 1 − �n,γ + 1) such that d(B; Yn) = (c∗1, . . . ,c∗n,yn,1,yn,2, . . . ) with c∗n+1 6= yn,1, so that d(B; Yn) 6= d∗(B; γ + 1). Suppose d(B; Yn) �B d∗(B; γ + 1). There exists Yn+m ∈ (Yn,γ + 1) where m ≥ 1 such that d(B; Yn+m) = (c∗1, . . . ,c ∗ n+m,yn+m,1,yn+m,2, . . . ). Since d(B; Yn) �B d∗(B,γ + 1), then (yn,1 − c∗n+1) sgn(B[n + 1]) ≥ 1, implying that d(B; Yn) �B d(B; Yn+m). By Proposition 4.5, Yn > Yn+m which is a contradiction since Yn+m ∈ (Yn,γ + 1). Hence, if x < Yn, then d(B; x) ≺B d(B; Yn) ≺B d∗(B; γ+1). Definition 4. A sequence (d1,d2, . . . ) ∈A(B) satis- fies the lexicographic restriction if, for all k ∈ N∪{0}, d(σk(B); γ) �σk (B) σk(d1,d2, . . . ) ≺σk (B) d ∗(σk(B); γ + 1). Combining Corollary 4.5.1 and Proposition 4.7 yields the following proposition. Proposition 4.8. Let x ∈ [γ,γ + 1). Then d(B; x) satisfies the lexicographic restriction. We now show that the converse of Prop. 4.8 holds under some condition. To proceed, consider a sequence z = (z1,z2, . . . ) ∈A(B). For i ∈ N, we define z(i,j) = { (zi,zi+1, . . . ,zi+j) if j ∈ N∪{0} (zi,zi+1, . . . ) if j = ∞ and set z(i,j)σi−1(B) = j∑ k=0 zi+k B[i, i + k] , provided that the sum converges if j = ∞. For n ∈ N∪ {0}, let u(n) = d(σn(B); γ) and v(n) = d∗(σn(B); γ + 1). Lemma 4.9. Let w = (w1,w2, . . . ) ∈ A(B). If w satisfies the lexicographic restriction, then there are infinitely many n such that at least one of the two holds: (1.) B[n] > 0 and (w(1,n− 1) ◦u(n))B ≥ γ; (2.) B[n] < 0 and (w(1,n− 1) ◦v(n))B ≥ γ. Proof. Suppose w 6= d(B; γ). For the base of the induction, we set n = 0 and define w(1,−1) as the empty word. Then (w(1,−1) ◦u(0))B = γ. Likewise (w(1,−1)◦v(0))B = γ + 1 ≥ γ. Now, let m ∈ N∪{0}. CASE 1 Suppose B[m] > 0 and (w(1,m − 1) ◦ u(m))B ≥ γ hold. By the lexicographic restriction, u(m) = d(σm(B); γ) ≺σm(B) σm(w) = w(m + 1,∞). Thus, there exists a least positive integer l such that wm+i = u(m)(i, 0) for all i < l and (wm+l −u(m)(l, 0)) sgn(σm(B)[l]) ≥ 1. Since B[m] > 0, then sgn(σm(B)[l]) = sgn(B[m + l]). CASE 1.1 Suppose B[m + l] > 0 so that (wm+l − u(m)(l, 0)) ≥ 1. We have (w(1,m + l− 1) ◦u(m+l))B − (w(1,m− 1) ◦u(m))B = wm+l −u(m)(l, 0) B[m + l] + (u(m+l))σm+l(B) − (u(m)(l + 1,∞))σm+l(B) B[m + l] . Now, wm+l −u(m)(l, 0) B[m + l] ≥ 1 B[m + l] . Meanwhile, (u(m+l))σm+l(B) − (u(m)(l + 1,∞))σm+l(B) = γ −T lσm(B)(γ). Hence, (u(m+l))σm+l(B) − (u(m)(l + 1,∞))σm+l(B) B[m + l] is greater than −1/B[m + l] and less than or equal to 0. Therefore, (w(1,m+l−1)◦u(m+l))B−(w(1,m−1)◦u(m))B ≥ 0, implying that (w(1,m+l−1)◦u(m+l))B > (w(1,m−1)◦u(m))B ≥ γ. CASE 1.2 Suppose B[m + l] < 0. Then (w(1,m + l− 1) ◦v(m+l))B − (w(1,m− 1) ◦u(m))B = wm+l −u(m)(l, 0) B[m + l] + (v(m+l))σm+l(B) − (u(m)(l + 1,∞))σm+l(B) B[m + l] . Since B[m + l] < 0, we have wm+l −u(m)(l, 0) B[m + l] ≥ −1 B[m + l] . Moreover, (v(m+l))σm+l(B) − (u(m)(l + 1,∞))σm+l(B) = γ + 1 −T l σm(B)(γ). It follows that (v(m+l))σm+l(B) − (u(m)(l + 1,∞))σm+l(B) B[m + l] 222 vol. 60 no. 3/2020 Beta Cantor Series Expansion and Admissible Sequences is less than 0 but greater than or equal to 1/B[m + l]. Therefore, (w(1,m+l−1)◦v(m+l))B−(w(1,m−1)◦u(m))B ≥ 0, and consequently, (w(1,m+l−1)◦v(m+l))B > (w(1,m−1)◦u(m))B ≥ γ. CASE 2 Suppose B[m] < 0 and (w(1,m − 1) ◦ v(m))B ≥ γ hold. By the lexicographic restriction, σm(w) = w(m+1,∞) ≺σm(B) v(m) = d∗(σm(B); γ+1). Thus, there exists a least positive integer l such that wm+i = v(m)(i, 0) for all i < l and (v(m)(l, 0) −wm+l) sgn(σm(B)[l]) ≥ 1. Since B[m] < 0, we have sgn(σm(B)[l]) = −sgn(B[m + l]). As before, we have two subcases: sgn(B[m + l]) > 0 and sgn(B[m + l]) < 0. The proofs are similar. Analogous to Lemma 4.9, we have the following result. Lemma 4.10. Let w ∈A(B). If w satisfies the lexi- cographic restriction, then there are infinitely many n such that at least one of the two holds: (1.) B[n] < 0 and (w(1,n− 1) ◦u(n))B ≤ γ + 1; (2.) B[n] > 0 and (w(1,n− 1) ◦v(n))B ≤ γ + 1. We now apply Lemmas 4.9 and 4.10 to prove the next proposition. Proposition 4.11. Let w = (w1,w2, . . . ) ∈ A(B) such that the sum σk(w)σk (B) converges for all k ∈ N ∪{0}. If w satisfies the lexicographic restriction, then σk(w)σk (B) ∈ [γ,γ + 1). Proof. We show that γ ≤ wB ≤ γ + 1. Let En(w) := σn(w)σn(B), which by assumption converges. Then, for all n ∈ N, wB = n∑ k=1 wk B[k] + En(w) B[n] = w(1,n− 1)B + En(w) B[n] . Thus, as n tends to ∞, the quotient En(w)/B[n] tends to 0. For sufficiently large n, (w(1,n− 1) ◦u(n))B ≥ γ or (w(1,n− 1) ◦v(n))B ≥ γ by Lemma 4.9. So, wB − (w(1,n− 1) ◦ t(n))B = En(w) B[n] − (t(n))σn(B) B[n] = En(w) B[n] − C B[n] −→ 0 where (t(n),C) is either (u(n),γ) or (v(n),γ+1). There- fore, wB ≥ γ. In general, observe that as w satisfies the lexicographic restriction with respect to B, then σm(w) also satisfies the lexicographic restriction with respect to σm(B). Consequently, Lemmas 4.9 and 4.10 apply. In other words, letting σm(w), σm(B), (u(m))σm(B) and (v(m))σm(B) take the role of w, B, γ and γ + 1, respectively, in Lemma 4.9, we obtain the conclusion that σm(w)σm(B) ≥ γ. Likewise, we have σm(w)σm(B) ≤ γ + 1 for all m ∈ N ∪{0} by Lemma 4.10. The only thing we are left to do is to show that σm(w)σm(B) 6= γ + 1. Let z = (z1,z2, . . . ) denote the sequence d∗(σM−1(B); γ + 1). Let s be the least positive integer such that wM+i−1 = zi for 1 ≤ i < s and (zs −wM+s−1) sgn(σM−1(B)[s]) ≥ 1. Note that there exists Y ∈ [γ,γ + 1) such that d(σM−1(B); Y ) = (z1, . . . ,zs,ys+1,ys+2 . . . ). Then, |Y (s + 1,∞)σM +s−1(B) −w(M + s,∞)σM +s−1(B)| ≤ (γ + 1) −γ = 1. Therefore, Y −w(M,∞)σM −1(B) = (zs −wM+s−1) + Y (s + 1,∞)σM +s−1(B) σM−1(B)[s] − w(M + s,∞)σM +s−1(B) σM−1(B)[s] ≥ 1 + sgn(σM−1(B)[s])(Y (s + 1,∞)σM +s−1(B) |σM−1(B)[s]| − w(M + s,∞)σM +s−1(B)) |σM−1(B)[s]| ≥ 1 − 1 |σM−1(B)[s]| = 0. Since γ + 1 > Y ≥ w(M,∞)σM −1(B), then w(M,∞)σM −1(B) 6= γ + 1. In the previous proposition, an important part of the proof is the assumption that the sequence w = (w1,w2, . . . ) ∈A(B) has the property that the series σk(w)σk (B) converges for all k ∈ N ∪{0}. It is clear that if the base B = (β1,β2, . . . ) is eventually periodic, then this property holds for w. We can say more. First, note that the digits are bounded by uβi and vβi (see Section 2), which, in turn, satisfy max(|uβi|, |vβi|) ≤ (|βi| + 1)(|γ| + 1). Now, let us consider the following. For the base B, let |B| be the sequence (|β1|, |β2|, . . . ). Suppose that (|β1| + 1, |β2| + 1, . . . )|B| = ∞∑ n=1 |βn| + 1 |B[n]| < ∞. (??) 223 Jonathan Caalim, Shiela Demegillo Acta Polytechnica Then for every sequence w ∈ A(B), the sum wB is convergent. Indeed,∣∣∣∣∣ ∞∑ n=1 wn B[n] ∣∣∣∣∣ ≤ ∞∑ n=1 ∣∣∣∣ wnB[n] ∣∣∣∣ ≤ ∞∑ n=1 (|βn| + 1)(|γ| + 1) |B[n]| ≤ (|γ| + 1) ∞∑ n=1 |βn| + 1 |B[n]| < ∞. Note that if B is eventually periodic, then (??) holds. However, if B = (b1,b2, . . . ) with bn = (n+ 1)/n, then (??) does not hold. We now state the main result of this article, which provides a sufficient and necessary condition for ad- missibility of integer sequence in A(B) with respect to the beta Cantor series expansion for a base sequence B satisfying (??). It would be interesting to know how the result can be extended beyond property (??). Theorem 4.12. Let B ∈ RN such that limm→∞ |B[m]| = ∞ and (??) holds. Let (d1,d2, . . . ) ∈ A(B). Then (d1,d2, . . . ) is B- admissible if and only if (d1,d2, . . . ) satisfies the lexi- cographic restriction. Acknowledgements The authors would like to thank the anonymous reviewer for valuable remarks that improved the quality of the paper. J. Caalim is grateful to the University of the Philippines for the financial support through its PhD Incentive Award under the Office of the Vice Chancellor for Research and Development. S. Demegillo is grateful to the Department of Science and Technology of the Philippine Government for the financial support under the DOST-ASTHRDP scholarship grant. References [1] A. Rényi. Representations for real numbers and their ergodic properties. Acta Math Acad Sci Hungar 8:477–493, 1957. doi:10.1007/BF02020331. [2] W. Parry. On the β-expansions of real numbers. 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Journal of Integer Sequences 15:12.2.6/1 – 12.2.6/21, 2012. 224 http://dx.doi.org/10.1007/BF02020331 http://dx.doi.org/10.1007/BF02020954 http://dx.doi.org/10.1007/bf01897025 http://dx.doi.org/10.1007/978-3-642-02737-6_20 http://dx.doi.org/10.1515/INTEG.2009.023 http://dx.doi.org/10.1017/S0143385711000514 http://dx.doi.org/10.1016/j.tcs.2011.08.028 http://dx.doi.org/10.1017/etds.2012.127 http://dx.doi.org/10.2969/jmsj/06910397 http://dx.doi.org/10.1007/s00454-016-9849-4 Acta Polytechnica 60(3):214–224, 2020 1 Introduction 2 B-expansion maps 3 Expansion of +1 4 Admissible Sequences Acknowledgements References