Periodicity and pure periodicity in alternate base systems

: Alternate base B is given by a p -tuple ( β 1 , β 2 , . . . , β p ) of real numbers greater than 1. We investigate in which cases all rational numbers pq in the interval (0 , 1) have an eventually periodic B -expansion. We show that this property forces the product δ = β 1 β 2 · · · β p to be a Pisot or a Salem number. Analogic conclusion was earlier derived by Charlier, Cisternino and Kreczman, under a stronger requirement that pq has an eventually periodic expansion in every alternate base obtained by a cyclic shift of the original p -tuple. We further examine under which circumstances there exists a γ > 0 such that every rational number in the interval (0 , γ ) has a purely periodic B -expansion. We show that a necessary condition for this phenomenon is that δ is a Pisot or a Salem unit. We also provide a suﬃcient condition. We thus generalize the results known for the R´enyi numeration system, i.e. for the case when p = 1, obtained by Schmidt, Akiyama, Adamczewski et al. and others. At the end, we present a class of alternate bases with p = 2, for which γ can be chosen to be 1.


Introduction
Cantor real base systems were first studied by Caalim and Demglio in [5] and independently by Charlier and Cisternino in [6] as a generalisation of Rényi β-expansions [16].While in the Rényi numeration system, one uses for representation of numbers a sum of powers of a single base β > 1, here we consider a sequence of real bases B = (β i ) i≥1 , β i > 1.A real number x ∈ [0, 1) can be represented by an infinite series Note that the possibility to represent real numbers in this form was already mentioned in [11].Some of the properties of such representations are direct analogies of those proved for β-expansions, others appear to be much more difficult.Papers [5] and [6] concentrated on characterizing the representations which are produced by the greedy algorithm, the so-called B-expansions.The characterization is given in terms of a set of lexicographic conditions, which are to be compared to those obtained by Parry [15] for Rényi βexpansions.Charlier and Cisternino [6] then focused on the sequences of bases that are purely periodic with period of length p.They called such a base B an alternate base, and write B = (β 1 , . . ., β p ).They then characterized alternate bases providing sofic systems.Algebraic description of sofic alternate bases is given in [8].Note that for p = 1, one obtains the case of Rényi β-expansions where soficness was described by Bertrand-Mathis [4].
From the arithmetical point of view, one is interested which numbers have B-expansions with finite, purely periodic or eventually periodic B-expansions.The so-called finiteness property (F), i.e. the fact that addition and subtraction of finite B-expansions yields again a finite B-expansions, was studied in [14], providing some necessary and some sufficient conditions for finiteness, a counterpart of the results of Frougny and Solomyak [10] and others.A class of bases with (F) property was also given.
The purpose of this article is to study the set Per(B) of numbers in the unit interval [0, 1) with periodic expansions in alternate base numeration systems.For p = 1, Schmidt [17] has shown that if Per(β) contains all rational numbers of [0, 1), then β is a Pisot number or a Salem number.As a Theorem A. Let B = (β 1 , . . ., β p ) be an alternate base and set δ = p i=1 β i .If Q ∩ [0, 1) ⊆ Per(B), then δ is either a Pisot or a Salem number and β 1 , . . ., β p ∈ Q(δ).
The second part of our results concerns rational numbers with purely periodic B-expansions.We will say that an alternate base B satisfies pure periodicity property (Property (PP)), if there exists an interval [0, γ), 0 < γ ≤ 1, such that every rational in [0, γ) has purely periodic B-expansion.A non-trivial problem is determination of the supremum of all constants γ exhibiting Property (PP) in base B. Let us denote it by γ(B).
Before stating our results, let us recall what is known for the case when p = 1.For Rényi βexpansions, Schmidt has shown that quadratic Pisot units with minimal polynomial x 2 − mx − 1, m ≥ 1, satisfy (PP), moreover with γ(β) = 1.Later, Hama and Imahashi [12] derived that if β is a quadratic Pisot unit not of this type (i.e. with minimal polynomial x 2 − mx + 1, m ≥ 3), then no rational number has purely periodic β-expansion, thus β does not possess (PP).
Akiyama [2] has put Property (PP) into connection with the finiteness property.In particular, he proved the following.
The question whether validity of (F) is necessary for (PP) has been decided for quadratic bases β (as a result of Schmidt [17] and Hama and Imahashi [12]) and also for cubic bases β.This is a result of Adamczewski et al. [1] who prove that a cubic base β satisfies (PP) if and only if it is a Pisot unit with (F).Moreover, they show that the constant γ(β) from Property (PP) is irrational for cubic numbers which are not totally real.
In this paper we study Property (PP) of alternate bases.We show a necessary condition.
Theorem C is shown in Section 5.In the last Section 6 we give a class of alternate bases with (PP) for which the constant γ(B) is equal to 1.We also illustrate the fact that γ(B (i) ) may be different from γ(B).

Preliminaries
Cantor real base is given by a sequence B = (β k ) k≥1 of real numbers The sequence of integer digits The greatest B-representation of x in lexicographic order, called the B-expansion of x, is the one obtain by the greedy algorithm: Set r 0 = x, and for k ≥ 0 set a k+1 = ⌊β k+1 r k ⌋, r k+1 = β k+1 r k − a k+1 .We denote the B-expansion of For characterisation of integer sequences that are admissible as B-expansions of numbers from the interval [0, 1), one needs to define the quasigreedy expansion of 1, denoted d * B (1), as the lexicographically greatest B-representation of 1 with infinitely may non-zero digits.
By we denote the standard lexicographic order; w ω stands for infinite repetition of the string w.
If the base B is a purely periodic sequence with period length p, i.e. β k+p = β k for any k ≥ 1, then B (k+p) = B (k) for any k ≥ 1.In this case we speak about an alternate base and write B = (β 1 , . . ., β p ).The special case when p = 1 corresponds to the numeration system with a single base β > 1, as was defined by Rényi and extensively studied by many authors from very diverse points of view.
In [14] the set Fin(B) of numbers with B-expansions having only finitely many non-zero digits is considered.We call such expansions finite.We denote We say that the base B satisfies the finiteness property (F), if for any x, y ∈ Fin(B), we have In [14], some necessary and some sufficient conditions for an alternate base B with period p to satisfy (F) are presented.Among other, it is shown that if B satisfies the finiteness property, then δ = p i=1 β i is a Pisot or a Salem number, β i ∈ Q(δ) and for any non-identical embedding ψ of Q(δ) into C, the vector (ψ(β 1 ), . . ., ψ(β p )) is not positive.
Recall that a complex number δ > 1 is a Pisot number, if it is an algebraic integer, i.e. a root of a monic polynomial with integer coefficients, with all conjugates in the interior of the unit circle.The number δ > 1 is a Salem number, if it is an algebraic integer with all conjugates in the unit circle and at least one of modulus equal to 1.The algebraic extension of rational numbers by δ is denoted by where n is the degree of δ as an algebraic number.Such a field Q(δ) has n embeddings into C (including the identity), i.e. field monomorphisms ψ : Q(δ) → C, induced by δ → δ ′ where δ ′ is a conjugate of δ.
In this paper we are particularly interested in numbers with eventually and purely periodic Bexpansions.According to [7], we define Per(B) = {x ∈ [0, 1) : d B (x) is eventually periodic}.
The second part of this paper is focused to rational numbers with purely periodic B-expansion.Definition 4.An alternate base B = (β 1 , β 2 , . . ., β p ) has the Pure Periodicity Property (PP), if there exists γ > 0 such that d B (x) is purely periodic for every x ∈ [0, γ) ∩ Q.
In the proof of our result given in Theorem C, we will need to extend the definition of a B-expansion to numbers outside of the unit interval.In [9], this is done with full generality for a two-way Cantor real base, here we simplify the task by considering an alternate base B = (β 1 , . . ., β p ), with δ = p i=1 β i .For a given non-negative number Having as a convention that two expansions coincide if they are the same up to leading zeros, d B (x) is unique not dependent on the choice of k.With this in hand, we can define the set which gives the set of all real numbers whose absolute value has a finite B-expansion.Property (F) then translates to saying that fin(B) is closed under addition and subtraction.
We further define the B-integers as numbers having only zeros on the right from the fractional point.We denote

Proof of Theorem A
The expansion of a real number x ∈ [0, 1) in the alternate base The latter can be viewed as a representation of x in the base δ = p i=1 β i with digits d k belonging to the alphabet In order to simplify the notation, denote Then we can express the alphabet D as Suppose now that the B-expansion of x is eventually periodic.The lengths of the preperiod and the period can always be assumed to be multiples of p, say For the value of x we can therefore write Realizing that the digits d k ∈ D are of the form we can rewrite the value of the product xδ r (δ s − 1) as where g i , f i , i = 1, . . ., p, are polynomials with integer non-negative coefficients In the particular case where the B-expansion of x is purely periodic, we have r = 0, the polynomials g i for i = 1, . . ., p vanish and we can simplify to Now assume that we have p rational numbers p j q j , j = 1, . . ., p, with eventually periodic Bexpansions d B ( Without loss of generality, we can assume that all the expansions have the preperiod pr and the period ps of the same length, i.e. Then we have p equalities of the form (6) that can be rewritten together into a matrix form, M v = 0, where the matrix M is given by p (δ) − p 1 δ r (δ s −1) where for simplicity we have denoted Lemma 5. Let B = (β 1 , . . ., β p ) be an alternate base and δ = p i=1 β i .Suppose there exists a nonsingular p×p matrix M (X) whose entries are integer polynomials, i.e. belong to Z[X].Let M (δ) v = 0, where v is given by (4).Then the following hold.
Proof.By assumption, the determinant of M (X) is a non-zero polynomial with integer coefficients, say det M (X) = F (X).For Item 1), it suffices to realize that v is non-zero, and thus the matrix M (δ) must be singular.We have det M (δ) = 0 = F (δ), which proves that δ is an algebraic number.
Let us prove Item 2).Since M (δ) v = 0, the vector v is an eigenvector of M (δ) corresponding to the eigenvalue 0. As rank of M (δ) is p − 1, the corresponding eigenvector is unique up to multiplication by a constant.In particular, for any real vector u satisfying M (δ) u = 0 there exists α ∈ R such that u = α v.
Since the entries of M (δ) belong to Q(δ), we can choose the eigenvector u to have entries in Q(δ).We then have , for i = 2, 3, . . ., p, and In what follows, we will set the choice of the rational numbers p j q j so that the matrix M of (9) satisfies the assumptions of Lemma 5. Lemma 6.Let B = (β 1 , β 2 , . . ., β p ) be an alternate base.Suppose that there exists a constant γ > 0 such that every rational number in [0, γ) has eventually periodic B-expansion.Then

, p, and
Proof.We will first make a suitable choice of p rational numbers x (j) = p j q j , j = 1, 2, . . ., p from the interval [0, γ) and form a matrix M of the form (9). For the proof of the statement, we than use Lemma 5.
Fix positive m ∈ N and n ∈ N such that 1 δ n < γ.For every j = 1, 2, . . ., p denote Since the ordering of numbers in the interval (0, 1) corresponds to lexicographic order of their Bexpansions, the intervals I j are mutually disjoint.For each j = 1, 2, . . ., p choose a rational number x (j) = p j q j from the interval 1 δ n I j .The B-expansion of x is then of the form Thanks to the choice of n, each interval 1 δ n I j is a subset of ⊂ [0, γ), and therefore d B (x (j) ) is eventually periodic.Without loss of generality we assume that the preperiod of d B (x (j) ) is of length pr > p(n+m) and the period is of length ps > 0. Thus d B (x (j) ) is of the form (8) and we have a matrix equation M v = 0 for a matrix M as in (9), where for i, j = 1, . . ., p we have and the polynomials f (j) i and g (j) i are as in (10).In order to show that δ is an algebraic number, by Lemma 5, it suffices to verify that the determinant of M is equal to det M = F (δ) for some non-zero polynomial F ∈ Z[X].For that, it suffices to show that in each row of the matrix M (X), the degree of the polynomial at the diagonal is strictly larger than the degrees of polynomials at other positions in the row.
Since for every j = 1, . . ., p the digits x , we can derive that For the degrees of the polynomials h (j) i we therefore have deg h Moreover, the polynomial h i is monic for i = 1, . . ., p, and its degree is strictly larger than the degree of polynomials h (j) i , i = j.Formula for computation of the determinant of the matrix M ensures that det M = F has the same leading coefficient as the product of polynomials on the diagonal of M (δ), .
Thus deg F = (s + r − n − 1)(p − 1) + s + r and the leading coefficient of F is −q 1 q 2 . . .q p−1 p p .The number δ is therefore a root of a non-zero polynomial F ∈ Z[X] and hence δ is an algebraic number.
Let us now demonstrate Item 2) of the statement.By Lemma 5, it suffices to show that M (δ) is equal to p − 1.We will ensure this fact by choosing a suitable parameter m ∈ N. Let us stress that so-far our considerations used arbitrary positive integer m ∈ N.
of the matrix M is strictly diagonally dominant and thus it is non-singular.For the j-th row, we need to verify that From ( 10) and ( 13), and the fact that coefficients of all polynomials are non-negative, we can deduce the following estimates on f i (δ).We have and It is therefore sufficient to show that the right hand-sides of ( 16) and ( 17) satisfy or equivalently It is obvious that the latter is satisfied for sufficiently large m.Consequently, ( 15) is true, the submatrix ( 14) is non-sigular and therefore using Item 2) of Lemma 5, β i ∈ Q(δ) for i = 1, . . ., p.
For the proof of this lemma, we will use the following statement which follows from the prime number theorem, see [13, p. 494].
Proof of Lemma 7. In the proof of Lemma 6 we set for x (j) = p j q j any rational numbers from the interval 1  δ n I j .With this choice we have derived that δ is a root of a non-zero integer polynomial F with leading coefficient equal to −q 1 q 2 • • • q p p p .The only condition on the fixed index n ∈ N was that 1 δ n < γ.Now we show that a more meticulous choice of n ensures that the interval 1 δ n I j contains two fractions x (j) = 1 q j and x(j) = 1 qj , where q j and qj are mutually distinct primes.With this choice of two p-tuples of rational numbers, we obtain by the same procedure as before two polynomials with root δ, say F and F from Z[X], the first one with leading coefficient −q 1 q 2 • • • q p , the second one with leading coefficient −q 1 q2 • • • qp .Since I j are mutually disjoint, q 1 , . . ., q p , q1 , . . ., qp are distinct primes.Hence, the leading coefficients of F and F are coprime.By Bézout's lemma, there exists a monic polynomial with integer coefficients with root δ, and thus δ is an algebraic integer.
For the proof of Lemma 7 it is therefore sufficient to demonstrate how to find an integer n ∈ N so that each of the intervals 1 δ n I j , j = 1, . . ., p, contains two fractions 1 q j and 1 qj , such that q j and qj are distinct primes.For this, we use Proposition 8 stated above.
Denote by ℓ j and r j the left and right end-points of the interval I j defined in (11), respectively.Since in the lexicographic order the prefix 0 j 10 pm−j−1 defining the interval I j+1 is smaller than the corresponding prefix for I j , obviously, To such ε we find by Proposition 8 the number A 1 .Now we choose n ∈ N such that besides the inequality 1 δ n < γ we also have has its left end-point larger than A 1 and the ratio of its right and left end-points satisfies r j ℓ j ≥ (1 + ε) 2 .Proposition 8 ensures that K j contains two distinct primes, say q j and qj .Since q j , qj ∈ K j we have 1 q j , 1 qj ∈ ℓ j δ n , r j δ n = 1 δ n I j as we wanted to show.Lemma 9. Let B = (β 1 , β 2 , . . ., β p ) be an alternate base.Suppose that there exists a constant γ > 0 such that every rational number in [0, γ) has eventually periodic B-expansion.Then δ = p i=1 β i is a Pisot or a Salem number.
Proof.By Lemma 7, δ is an algebraic integer.It remains to show that no algebraic conjugate of δ is in modulus strictly larger than 1, i.e. δ is a Pisot or a Salem number.Assume for the contradiction that there exists an algebraic conjugate η of δ distinct from δ, which has modulus |η| > 1.
Fix now n ∈ N so that 1 δ n−1 < γ and δ n = η n .Then for every positive integer m the set m) is eventually periodic and has the form As the expansion is eventually periodic, we can apply to the latter relation the field isomorphism ψ induced by ψ(δ) = η to obtain Since the digit set D is finite, there exist constants K and K such that |d| ≤ K and |ψ(d)| ≤ K for every d ∈ D. The infinite sums in ( 18) and ( 19) can therefore be bounded by Subtracting ( 18) and ( 19) with the use of these estimates, we derive for every sufficiently large m ∈ N. The right hand side of the inequality tends to 0 with m increasing to the infinity, whereas the left hand side is a positive number independent of m.Hence by contradiction, a conjugate η of δ in modulus greater than 1 cannot exist.
Proof of Theorem A. It suffices to combine statements of Lemmas 6 and 9.

Proof of Theorem B
Theorem B gives a necessary condition for an alternate base B so that it satisfied Property (PP).Its proof is an adjustment of arguments used by Akiyama [2] for the case of alternate bases.
Proof.Assume that the B-expansion of every rational number in [0, γ) is purely periodic.From Lemma 7, we know that δ = p i=1 β i > 1 is an algebraic integer, say of degree D, and thus its norm Norm(δ) is an integer.Since β i ∈ Q(δ), and the alphabet D ⊂ Q(δ) is finite, there exists a positive integer q such that every digit d ∈ D can be written in the form In order to show that δ is an algebraic unit, we need to verify that Norm(δ) = ±1.Assume the contrary, i.e. ∆ := |Norm(δ)| ≥ 2. This allows us to find an n ∈ N such that the constant γ > 0 from the definition of Property (PP) satisfies for some s ∈ N. By (7), we have i.e. we have for some F ∈ Z[X] of degree deg(F ) ≤ s − 2 that x(δ s − 1) = 1 q F (δ). Substituting the value of x, the latter implies that δ s − 1 = ∆ n F (δ).In other words, δ is a root of the polynomial The constant term of the polynomial is equal to −1 mod ∆.As the norm of a root of any integer polynomial divides its constant term, we derive that ∆ divides −1 and hence ∆ = |Norm(δ)| = 1, which is a contradiction, proving Item 1).
Applying ψ on (20) gives This is a contradiction, since the number on the left hand-side is negative, whereas the right hand-side is ≥ 0.
Proof of Theorem B. It suffices to combine the statements of Lemma 6, Lemma 9 and Lemma 10.

Proof of Theorem C
Theorem C expresses a sufficient condition for an alternate base B to satisfy Property (PP).Two auxiliary statements are needed.
Lemma 11.Let δ > 1 be a Pisot unit and D denote the alphabet defined in (3).Assume that β 1 , β 2 , . . ., β p ∈ Q(δ).Then for every c > 0 there exists k ∈ N with the following property: Proof.Denote by D the degree of the algebraic number δ.The following facts hold true: Fact 1) Since δ is an unit, δ −i belongs to the ring , and thus there exists We prove the statement of the lemma by contradiction.Assume that there exists a constant c > 0 such that for any k ∈ N we can find a number y (k) with the B-expansion of the form ( Since δ is Pisot and 1 ∈ D, obviously Φ ∈ (0, 1) and µ ≥ 1.
For every z ∈ N B and for every non-identical embedding φ we have the following inequalities Remark 13.By [14], requirement that Property (F) is satisfied forces d B (i) (1) to be finite for every i = 1, 2, . . ., p.There exists only finitely many factors of the form a0 ℓ b with a, b = 0, ℓ ∈ N, belonging to the language of the string d B (i) (1) for some i.Choose r ∈ N so that ℓ < rp for every such factor a0 ℓ b.This choice of r ensures that if both x 1 x 2 • • • x pk 0 ω and y 1 y 2 y 3 • • • are B-admissible, then also Proof of Theorem C. We first show that B = B (1) satisfies Property (PP).
Let c > 0 denote the constant from Lemma 12 and let r ∈ N be as chosen in Remark 13.Set γ = min{c, δ −r }.
Let x ∈ Q ∩ (0, γ).We will use the fact that there exist infinitely many N ∈ N such that x(δ N − 1) ∈ Z[δ], see [2].Among such exponents, choose N so that |φ(δ)| • |φ(δ) N − 1| < 1, for every non-identical embedding φ.This is possible, because δ is a Pisot number, i.e. |φ(δ) Since B satisfies property (F), the B-expansion of z is of the form z = n j=s d j δ j , where s, n ∈ Z, s ≤ n, d s , d s+1 , . . ., d n ∈ D and d s = 0. We first show using Lemma 12 that z ∈ N B , in other words that s ≥ 0. Indeed, as δ −s z ∈ N B \δ N B , we can find a non-identical embedding ψ such that By Remark 13, the purely periodic string (0 rp z rp+1 z rp+1 • • • z pN ) ω is B-admissible.Moreover, the string represents the number In order to show that B (i) satisfies (PP) for i ≥ 2, it suffices to recall the result of [14] which states that if B satisfies the finiteness property, then so do all of its shifts.

A class of bases with (PP)
In this section we present a class of alternate bases in which any rational number in the interval [0, 1) has purely periodic expansion.), which is a shift of the example well studied in [6].
Proof.The number δ is a quadratic Pisot unit of the class studied in [17] for which Schmidt proved that the constant γ from the (PP) property is equal to 1. Thus every rational number r ∈ (0, 1) has a purely periodic expansion in the Rényi numeration system with base δ, i.e.
where, by the Parry condition [15], for each i ∈ N, . In particular, the digits d i belong to {0, 1, . . ., m + 1}, and each digit d i = (m + 1) is followed by the digit d i+1 = 0. We thus have We now justify that the last digit in the period is d n = 0. Indeed, from (23) we derive that Apply the non-identical automorphism ψ of the field Q(δ), which is induced by ψ Note that this B-representation of r need not be the B-expansion of r, for: (i) it may contain blocks 0(m+1) which do not represent digits of the alphabet D, (ii) the order of blocks ab does not respect the condition of admissibility of B-expansions (Theorem 3) given by ( 22).
We will define an algorithm which rewrites the representation (24) into an admissible B-expansion, by dealing with problems (i) and (ii).We will use simple relations which hold for β 2 and δ and can be derived straightforwardly from their definition, namely (i) Since d i = (m+1) implies d i+1 = 0, each block 0(m+1) appears in the sequence (24) in the pair 0(m+1) 00 .Rewrite 0(m+1) 00 → 10 10 The two strings represent the same value, thanks to (25).Use this rewriting rule for the string z = 0d 1 0d 2 • • • 0d n−1 00 of (24) until we have a representation of z as a string of blocks 0b , b ∈ {0, . . ., m}, and blocks 10 10 .Moreover, the string of blocks representing z ends either in 00 or in 10 10 .We will say that it is reduced.
(ii) From the condition of admissibility of B-expansions (Theorem 3) and the expansions of 1 in the considered base (22), we derive that the reduced sequence of blocks obtained in step (i) may contain non-admissible substrings only of the form 0m 0b , 1 ≤ b ≤ m (say Type A) or 0m 10 10 (Type B).For them, we can use the following rewriting rules: 0m 0b → 10 00(b−1) (28) and 0m 10 10 → 10 0m 00 .
One can verify using ( 26) and ( 27) that these rewriting rules preserve the value of the string.
We proceed from left to right by induction on the position of the left-most occurrence of a forbidden substring.In the reduced sequence of blocks we find the left-most occurrence of a forbidden string of Type A or Type B. Using the rewriting rules (28) or (29), respectively, we obtain again a reduced sequence of blocks whose left-most occurrence of a forbidden string is smaller by two blocks.
After finitely many steps, we obtain a reduced sequence of blocks which does not contain any forbidden strings, i.e. is admissible in base B.
At last, realize that step (ii) produces a B-admissible sequence of blocks representing the number z which ends with the block 00 , or the block 10 .In order to obtain the B-expansion of r, we need to concatenate the string for z infinitely many times.The condition on the ending block of z ensures that such a concatenation remains B-admissible.

Proposition 14 .
Let δ > 1 be the positive root of the polynomial x 2 − (m + 1)x − 1, m ≥ 1, and letβ 1 = δ δ−1 , β 2 = δ − 1.Then every rational number in the interval [0, 1) has purely periodic expansion, i.e. γ(B) = 1.It can be easily computed that⌊β 1 ⌋ = 1, ⌊β 2 ⌋ = m and d B (1) = 11, d B (2) (1) = m01, which implies d * B (1) = (10) ω , d * B (2) (1) = m0(01) ω .(22)Note that for m = 2, we obtain B = ( This gives that d n < 1, whence d n = 0. Let us now inspect the expansion of the rational number r in base B. In (2), we have shown that a representation of a number x in an alternate base B can be viewed as a representation of x in a single base δ with digits in the alphabet D. In our case, D = {aβ 2 + b : a, b ∈ N, a ≤ 1, b ≤ m}.Replacing the digits aβ 2 + b ∈ D by the pair ab , one converts the representation of a number x in base δ into a representation of x in the alternate base B. The Rényi expansion (23) of r is purely periodic with the period being the number z