aa r X i v : . [ m a t h . N T ] F e b FIBONACCI EXPANSIONS
CLAUDIO BAIOCCHI, VILMOS KOMORNIK, AND PAOLA LORETI
Abstract.
Expansions in the Golden ratio base have been studied since apioneering paper of R´enyi more than sixty years ago. We introduce closelyrelated expansions of a new type, based on the Fibonacci sequence, and weshow that in some sense they behave better. Introduction
Expansions of the form x = ∞ X i =1 c i q i , ( c i ) ∈ { , } N in real bases q ∈ (1 ,
2) have been first studied by R´enyi [20]. Subsequently manyworks have been devoted to their connections to probability and ergodic theory,combinatorics, symbolic dynamics, measure theory, topology and number theory;see, e.g., [11, 19, 18, 21, 3, 4, 15, 2, 10] and their references. Following the discoveryof surprising uniqueness phenomena by Erd˝os et al. [7], a rich theory of uniqueexpansions has been uncovered [8, 5, 9, 16, 17, 6] . There are still many openproblems, for example concerning the number of possible expansions of specificnumbers in particular bases.In the special case where q = ϕ := √ ≈ .
618 is the Golden ratio, R´enyiproved that the average distribution of the digits 0 and 1 is not the same, and hecomputed their frequencies. In this paper we introduce the
Fibonacci expansions x = ∞ X i =1 c i F i , ( c i ) ∈ { , } N , where the powers ϕ i are replaced by the Fibonacci numbers:(1.1) F := 1 , F := 1 , and F i +2 := F i +1 + F i , i = 1 , , . . . . They are closely related to the expansions in base ϕ , because(1.2) F i = 1 √ (cid:18) ϕ i + ( − i +1 ϕ i (cid:19) for all i by Binet’s formula, whence F i is the nearest integer to ϕ i / √ Mathematics Subject Classification.
Primary: 11A63, Secondary: 11B39.
Key words and phrases.
Non-integer base expansions, Kakeya sequences, Fibonacci sequence,Golden ratio.Claudio Baiocchi passed away on December 14, 2020. His co-authors are grateful to his essentialcontributions to our collaboration.The work of the second author was partially supported by the National Natural Science Foun-dation of China (NSFC)
The purpose of this work is to compare these two expansions, and to study moregeneral
Kakeya expansions of the form(1.3) x = ∞ X i =1 c i p i , ( c i ) ∈ { , } N , where ( p i ) is a given Kakeya sequence , i.e., a sequence of positive numbers satisfyingthe conditions p i →
0, and(1.4) p n ≤ ∞ X i = n +1 p i for all n. We recall the following classical theorem:
Theorem 1.1 (Kakeya [12, 13]) . If ( p i ) is a Kakeya sequence, then a real number x has an expansion of the form (1.3) if and only if x ∈ [0 , P ∞ i =1 p i ] . For example, ( q − i ) is a Kakeya sequence for every q ∈ (1 , x ∈ [0 , q − ] has an expansion in base q . A similar result holds for Fibonacciexpansions. Setting(1.5) S := ∞ X i =1 F i ≈ . , we have the following result: Theorem 1.2.
A real number x has an expansion in the Fibonacci base if and onlyif x ∈ [0 , S ] . Next we will investigate the number of expansions. It is clear that the expansionsof 0 and q − are unique in every base q ∈ (1 , c i ≡ c i ≡
1, respectively.Otherwise, x may have several, even infinitely many expansions: Theorem 1.3 (Erd˝os et al. [8]) . If q ∈ (1 , ϕ ) , then every x ∈ (0 , q − ) has acontinuum of expansions in base q . By a different proof, we will extend Theorem 1.3 to a class of Kakeya expansions:
Theorem 1.4.
Let ( p i ) be a sequence of positive real numbers, satisfying the fol-lowing conditions: p i → p n < ∞ X i = n +1 p i for all n ;(1.7) p n − < ∞ X i = n +1 p i for infinitely many n ;(1.8) p n ≤ p n +1 for all sufficiently large n. (1.9) Then every < x < S := P ∞ i =1 p i has a continuum of expansions of the form (1.3) . The assumption q ∈ (1 , ϕ ) of Theorem 1.3 is sharp: if q = ϕ , then for example1 has only countably many expansions by a theorem of Erd˝os et al. [7]. ApplyingTheorem 1.4 we will prove that the Fibonacci expansions behave better: Theorem 1.5.
Every x ∈ (0 , S ) has a continuum of Fibonacci expansions. IBONACCI EXPANSIONS 3
For the reader’s convenience we give a short proof of Theorem 1.1 in Section 2.Then Theorems 1.2, 1.4 and 1.5 are proved in Sections 3, 4 and 5 , respectively.Ate the end of Section 4 we also deduce Theorem 1.3 from Theorem 1.4.2.
Proof of Theorem 1.1
First we prove Theorem 1.1. Given an arbitrary x ∈ [0 , S ], we define a function f : N → { , } recursively as follows. Set s := x and s := S − x . If n ≥ f (1) , . . . , f ( n −
1) have already been defined (no assumption if n = 1), then wechoose j ∈ { , } such that p n + X i
We need a lemma:
Lemma 3.1.
We have F i → + ∞ . Furthermore, F n +1 ≤ F n and F n < ∞ X i = n +1 F i for all n .Proof. It follows from the definition that the Fibonacci sequence is a strictly in-creasing sequence of positive integers. Therefore F i → + ∞ .Since F < F and F = 2 F , a trivial induction argument based on the identity F i +2 = F i +1 + F i shows that F n +1 ≤ F n for all n ≥
1, and equality holds onlyif n = 2. From this, again by induction, we have F n + j ≤ j F n for all n ≥ j ≥
1, and equality holds only if n = 2 and j = 1. Indeed, we have F n + j ≤ F n + j − ≤ F n + j − ≤ · · · ≤ j F n , CLAUDIO BAIOCCHI, VILMOS KOMORNIK, AND PAOLA LORETI and at least one the inequalities is strict unless n = 2 and j = 1. Therefore ∞ X j =1 F n + j > ∞ X j =1 j F n = 1 F n for every n . (cid:3) Proof of Theorem 1.2.
Since the Fibonacci numbers are positive, the formula p i :=1 /F i defines a Kakeya sequence by Lemma 3.1, and Theorem 1.2 follows fromTheorem 1.1. (cid:3) We may place Theorem 1.2 into a broader framework, by considering expansionsof the form(3.1) x = ∞ X i =1 c i q i (1 + ε i ) , ( c i ) ∈ { , } N with some given real numbers q ∈ (1 ,
2) and ε i > −
1. The following result showsthat ( q − i ), and even some perturbations of ( q − i ) are Kakeya sequences. Proposition 3.2. If < q < and (3.2) 1 + inf j ε j j ε j ≥ q − , then (cid:16) q i (1+ ε i ) (cid:17) is a Kakeya sequence.Proof. By our assumption we have q i (1 + ε i ) > i , and 1 + inf j ε j > q i (1 + ε i ) ≥ q i (1 + inf j ε j ) → + ∞ , and therefore q i (1+ ε i ) → n ≥ q n (1 + ε n ) ≤ ∞ X i = n +1 q i (1 + ε i ) . They follow from the relations1 q n (1 + ε n ) ≤ q n (1 + inf j ε j ) ≤ ∞ X i = n +1 q i (1 + sup j ε j ) ≤ ∞ X i = n +1 q i (1 + ε i ) , where the first and third inequalities ar obvious, while the middle inequality isequivalent to 1 + sup j ε j j ε j ≤ q − , i.e., to our assumption (3.2). (cid:3) Example.
By Binet’s formula (1.2) the Fibonacci expansions are equivalent to theexpansions (3.1) with q = ϕ and ε i = ( − i +1 ϕ i . In this case we haveinf j ε j = ε = − ϕ and sup j ε j = ε = 1 ϕ . IBONACCI EXPANSIONS 5
Hence 1 + inf j ε j j ε j = 1 − ϕ ϕ = ϕ − ϕ ( ϕ + 1) = ϕ − ϕ = ϕ − ϕ + 1 = ϕ − , so that the condition of Proposition 3.2 is satisfied.4. Proof of Theorem 1.4
We need a new lemma. Let ( p i ) be as in Theorem 1.4. We say that p n is a special element if the condition (1.8) is satisfied. Lemma 4.1.
There exists a K ≥ such that if we remove from ( p i ) a specialelement p k with k > K , then the remaining sequence still satisfies the correspondinghypotheses of Theorem 1.4.Proof. Fix an N ≥ p n ≤ p n +1 for all n ≥ N, and set ε := min ( − p n + ∞ X i = n +1 p i : n = 1 , . . . , N ) . By assumption (1.7) we have ε >
0. Since p i →
0, there exists a K such that p i < ε for all i ≥ K .If we remove from ( p i ) a special element p k with k ≥ K , then the conditions(1.6), (1.8) and (1.9) obviously remain valid. The condition (1.7) also remains validfor all n > k because the corresponding inequalities are unchanged, and it alsoremains valid for all n ≤ N by the choice of K . It remains to show that(4.1) p n + p k < ∞ X i = n +1 p i for all N < n < k.
This is true for n = k − p k is a special element. Proceeding byinduction, if (4.1) holds for some N < n < k , then it also holds for n −
1. Indeed,since p n − ≤ p n by our choice of N , applying (4.1) we get p n − + p k ≤ p n + p k < p n + ∞ X i = n +1 p i = ∞ X i = n p i . (cid:3) Proof of Theorem 1.4.
Given 0 < x < S , using (1.6) we may apply repeatedlyLemma 4.1 to construct a sequence p i > p i > · · · of special elements such that(4.2) ∞ X j =1 p i j ≤ min { x, S − x } . We may assume that i j +1 > i j + 1 for infinitely many indices j ; then after theremoval of the elements p i j we still have an infinite sequence, that we denote by( p ′ i ). Since after the removal of any finite number of elements p i , . . . , p i m theremaining sequence still satisfies the corresponding conditions (1.6) and (1.7) ofTheorem 1.4, letting m → ∞ we conclude that ( p ′ i ) is a Kakeya sequence. Now we CLAUDIO BAIOCCHI, VILMOS KOMORNIK, AND PAOLA LORETI may obtain a continuum of expansions of x as follows. Fix an arbitrary sequence( c i j ) ⊂ { , } . Then 0 ≤ ∞ X j =1 c i j p i j ≤ min { x, S − x } by (4.2), so that 0 ≤ x − ∞ X j =1 c i j p i j ≤ x ≤ S − ∞ X j =1 p i j = ∞ X i =1 p ′ i . By Theorem 1.1 there exists a sequence ( c ′ i ) ⊂ { , } such that ∞ X i =1 c ′ i p ′ i = x − ∞ X j =1 c i j p i j . Then ∞ X j =1 c i j p i j + ∞ X i =1 c ′ i p ′ i is an expansion of x in the original system ( p i ) (the order of the positive terms isirrelevant), and different sequences ( c i j ) lead to different expansions. (cid:3) In order to state a corollary of Theorem 1.4 we generalize the geometric se-quences. Given a real number ρ >
0, a sequence ( p i ) of positive numbers is calleda ρ -sequence if p i +1 ≥ ρ · p i for all i. Corollary 4.2. If ( p i ) is a ρ -sequence with ρ > ϕ , and p i → , then every x ∈ (0 , P i p i ) has a continuum of expansions x = ∞ X i =1 c i p i , ( c i ) ∈ { , } N . Proof.
It suffices to check the conditions of Theorem 1.4. We have p i → p n ≤ ρ p n +1 < ϕp n +1 < p n +1 for all n , so that the assumptions (1.6) and (1.9) are satisfied. Furthermore, since( p i ) is a ρ -sequence with ρ > ϕ , and since ρ < p i →
0, thefollowing relations hold for every n ≥ ∞ X i = n +1 p i ≥ p n − ∞ X i =2 ρ i = ρ − ρ p n − > p n − . This proves (1.8) for all n ≥
2, and this implies (1.7). (cid:3)
Example.
Corollary 4.2 reduces to Theorem 1.3 if p i = q − i with q ∈ (1 , ϕ ). IBONACCI EXPANSIONS 7 Proof of Theorem 1.5
For the proof of Theorem 1.5 we need some more properties of the Fibonaccisequence. We recall the identity(5.1) F i = F i − F i +1 + ( − i +1 , i = 2 , , . . . . It holds for i = 2 by a direct inspection: 1 = 1 · −
1. Proceeding by induction, ifit holds for some i ≥
2, then it also holds for i + 1 because F i = F i − F i +1 + ( − i +1 = ⇒ F i ( F i + F i +1 ) = ( F i − + F i ) F i +1 + ( − i +1 = ⇒ F i F i +2 = F i +1 + ( − i +1 = ⇒ F i +1 = F i F i +2 + ( − i +2 . Lemma 5.1. If k is a positive odd integer, then (5.2) 1 F k < ∞ X i = k +2 F i . Proof.
Fix any positive integer k , and set α := F k +1 F k , β := F k +2 F k +1 , γ := max { α, β } . Then F i +1 ≤ γF i for i = k, k + 1, and hence by induction for all i ≥ k . Since α = β by (5.1), one of the equalities F k +1 ≤ γF k and F k +2 ≤ γF k +1 is strict. Therefore ∞ X i = k +2 F i > F k +1 ∞ X j =1 γ j = 1 F k +1 · γ − F k · α · ( γ − . If k is odd, then this implies (5.2) because β > α by (5.1), so that γ = β , and hence α · ( γ −
1) = F k +1 F k · (cid:18) F k +2 F k +1 − (cid:19) = F k +2 − F k +1 F k = 1 . (cid:3) Proof of Theorem 1.5.
The conditions (1.6), (1.7) et (1.9) of Theorem 1.4 are sat-isfied for p i := 1 /F i by Lemma 3.1, and (1.8) is satisfied by Lemma 5.1. (cid:3) Acknowledgment.
The authors thank Mike Keane for suggesting the study ofFibonacci expansions.
References [1] R. Alcaraz Barrera, S. Baker, D. Kong,
Entropy, topological transitivity, and dimensionalproperties of unique q-expansions,
Trans. Amer. Math. Soc. 371(5) (2019), 3209–3258.[2] S. Akiyama, V. Komornik,
Discrete spectra and Pisot numbers . J. Number Theory 133 (2013),no. 2, 375–390.[3] C. Baiocchi, V. Komornik,
Greedy and quasi-greedy expansions in noninteger bases , arXiv:math/0710.3001.[4] K. Dajani, M. de Vries,
Invariant densities for random β -expansions , J. Eur. Math. Soc. 9(2007), no. 1, 157–176.[5] Z. Dar´oczy, I. K´atai, On the structure of univoque numbers , Publ. Math. Debrecen 46 (1995),3–4, 385–408.[6] M. de Vries, V. Komornik,
Unique expansions of real numbers , Adv. Math. 221 (2009), 390–427.[7] P. Erd˝os, M. Horv´ath, I. Jo´o, On the uniqueness of the expansions 1 = P q − n i . Acta Math.Hungar. (1991) 333–342. CLAUDIO BAIOCCHI, VILMOS KOMORNIK, AND PAOLA LORETI [8] P. Erd˝os, I. Jo´o, V. Komornik, Characterization of the unique expansions 1 = P q − n i andrelated problems. Bull. Soc. Math. France (1990) 377–390.[9] P. Erd˝os, V. Komornik,
Developments in noninteger bases , Acta Math. Hungar. 79 (1998),no. 1–2, 57–83.[10] De-Jun Feng,
On the topology of polynomials with bounded integer coefficients . J. Eur. Math.Soc. 18 (2016), no. 1, 181–193.[11] A. O. Gelfond,
A common property of number systems (Russian). Izv. Akad. Nauk SSSR.Ser. Mat. 23 (1959), 809–814.[12] S. Kakeya, On the set of partial sums of an infinite series.
Proc. Tokyo Math.-Phys. Soc. (2) (1914) 250–251.[13] S. Kakeya, On the partial sums of an infinite series. Tˆohoku Sc. Rep. (1915) 159–163.[14] V. Komornik, D. Kong, W. Li, Hausdorff dimension of univoque sets and devil’s staircase.
Adv. Math. 305 (2017), 165–196.[15] V. Komornik, A. C. Lai, M. Pedicini,
Generalized golden ratios of ternary alphabets . J. Eur.Math. Soc. 13 (2011), 4, 1113–1146.[16] V. Komornik, P. Loreti,
Unique developments in noninteger bases , Amer. Math. Monthly,105 (1998), 636–639.[17] V. Komornik, P. Loreti,
On the topological structure of univoque sets . J. Number Theory,122 (2007), 157–183.[18] V. Komornik, P. Loreti, M. Pedicini,
On an approximation property of Pisot numbers . J.Number Theory 80 (2000), 218–237.[19] W. Parry,
On the β -expansion of real numbers . Acta Math. Hungar. 11 (1960), 401–416.[20] A. R´enyi, Representations for real numbers and their ergodic properties. Acta Math. Hungar. (1957) 477–493.[21] N. Sidorov, Arithmetic dynamics . In
Topics in dynamics and ergodic theory , volume 310 of
London Math. Soc. Lecture Note Ser. , 145–189. Cambridge Univ. Press, Cambridge.
Accademia Nazionale dei Lincei, Palazzo Corsini, Via della Lungara 10, 00165 Roma,ItalyD´epartement de math´ematique, Universit´e de Strasbourg, 7 rue Ren´e Descartes,67084 Strasbourg Cedex, France
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