Multiple common expansions in non-integer bases

Acta Scientiarum Mathematicarum manuscript No. (will be inserted by the editor)

Multiple common expansions in non-integer bases

Vilmos Komornik Attila Peth®

·

Marco Pedicini

·

Version of November 10, 2015

Abstract In this paper we present results on the existence of simultaneous

representations of real numbers x in bases 1 < q1 < . . . < qr , r ≥ 2 with the digit set A = {−1, 0, 1}. We prove among others that each real number has innitely many common expansions. In general the base numbers depend on x. Restricting ourself to a small interval I we can choose at least one base independently of x ∈ I .

Keywords simultaneous Rényi expansions · interval lling sequences Mathematics Subject Classication (2000) 11A63 · 11B83 1 Introduction Given a nite alphabet or digit set A of real numbers and a real base q > 1, by an expansion of a real number x we mean a sequence c = (ci ) ∈ A∞ satisfying The third author was partially supported by the OTKA grants No. 100339 and 104208. V. Komornik Dèpartement de mathèmatique Universitè de Strasbourg 7 rue Renè Descartes 67084 Strasbourg Cedex, France E-mail: [email protected] M. Pedicini Dipartimento di Matematica e Fisica Università Roma Tre Via della Vasca Navale 84 00146 Roma, Italy E-mail: [email protected] A. Peth® Department of Computer Science University of Debrecen H4010 Debrecen P.O. Box 12 Hungary E-mail: [email protected]

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the equality

∞ X ci = x. i q i=1

This concept was introduced by Rényi [7] as a generalization of the familiar integer base expansions. Given a nite number of dierent bases q1 , . . . , qr , we may ask whether there exist real numbers having the same expansions in all these bases: ∞ ∞ X X ci ci = · · · = = x. i i q q i=1 1 i=1 r

(1)

The case r = 2 has been investigated in [5] for the alphabets A = {−m, . . . , 0, . . . , m}, m = 1, 2, . . . . In this case the trivial expansion ci ≡ 0 always

satises (1). We recall a special case of [5, Theorem 1]:

Theorem 1 If 1 < q1 ≤ m + 1 and q2 > q1 , then there exist innitely many sequences (ci ) ∈ {−1, 0, 1}∞ satisfying the equality ∞ ∞ X X ci ci = . i i q q i=1 2 i=1 1

The methods applied in the above paper did not allow us to construct common expansions for more than two bases. The purpose of this paper is to prove some results of this kind, by applying a dierent approach. We limit ourselves to the case m = 1, i.e., to the alphabet {−1, 0, 1}.1 Our rst result is the following:

Theorem 2 Given a positive integer r, there exists a non-empty open interval I such that for each x ∈ I there exist r real numbers 1 < q1 < · · · < qr and a sequence (ci ) ∈ {−1, 0, 1}∞ satisfying the equalities ∞ ∞ X X ci ci = · · · = = x. i qi q i=1 1 i=1 r

(2)

Compared to Theorem 1, the price to pay for dealing with more than two bases is that we cannot x them independently of the choice of x. The following improvement of Theorem 2 allows us to choose at least one base independently of x:

Theorem 3 Given a positive integer r and a real number two non-empty open intervals I, J satisfying 1 ∈ I,

p > 1, there exist

J ⊂ (1, p),

1 The results of the paper extend easily to the case of m > 1, the case m = 1 being the most dicult.

Multiple common expansions in non-integer bases

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and having the following properties: for any xed x ∈ I and q1 ∈ J there exist real numbers q2 , . . . , qr satisfying the inequalities q1 < · · · < qr < p,

and a sequence (ci ) ∈ {−1, 0, 1}∞ satisfying the equalities (2).

Finally, we prove the following improvement of Theorem 2 stating the existence of common expansions of each real number in innitely many bases:

Theorem 4 There exists a continuum of sequences

that for each real number x the equality

(ci ) ∈ {−1, 0, 1}

∞

such

∞ X ci qi i=1 j

holds for innitely many bases qj .

2 Proof of Theorem 3 We may assume without loss of generality that 1 < p < 3/2. Denoting by √ ϕ := (1 + 5)/2 ≈ 1.618 the golden ratio, we x a large positive integer n satisfying −n

(ϕ − 1)3

>

1 , p

and we introduce the polynomial n+r Y

f (t) := −

k=n+1



k

k

1 − t3 − t2·3



=

d X

ci ti .

i=0

We observe that c0 = f (0) = −1, and that all coecients ci belong to the set {−1, 0, 1}. Indeed, developing the product we obtain 3r terms of dierent degrees. Since y 2 + y − 1 = (y − ϕ + 1)(y + ϕ),

f has r real positive roots: −n−1 −n−r 1 < (ϕ − 1)3 < · · · < (ϕ − 1)3 < 1, p

and r real negative roots: −k

−ϕ3

,

k = n + 1, . . . , n + r.

Furthermore, since the exponents 3k are odd integers, all other roots of f are non-real, and all roots of f are simple.2 2 The roots of f are the vertices of 2r regular odd-sided polygons of dierent radii, centered at the origin, and each polygon has a unique vertex on the real axis.

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Write αk := (ϕ−1)3 the inequalities

−n−k

for brevity, and x r+1 real numbers βk satisfying

1 < β0 < α1 < β1 < α2 < · · · < βr−1 < αr < βr < 1. p

(3)

Since each root αk is simple, f (β0 ), . . . , f (βr ) have alternating signs.

(4)

Now set

1 min {|f (β0 )| , . . . , |f (βr )|} (> 0), 2 choose a large positive integer N ≥ d satisfying the inequality ε :=

∞ X

βri < ε,

i=N +1

and dene εN :=

∞  α i X r (< ε). 2

i=N +1

Since f is continuous and f (αr ) = 0, we may x a small positive number δ such that εN |t − αr | < δ =⇒ |f (t)| < . 2 We may assume that 0 1), it will follow that f takes each real value x innitely many times. Starting with n0 := 0, we are going to construct by recursion a sequence 0 = n0 < n1 < n2 < · · · of integers and a sequence p1 > p2 > · · · of real numbers, converging to one, as follows. First step. We choose p1 > 1 suciently small, e.g., p1 < 2, such that 1 p1 −1 > 1, then a large positive integer n1 such that n1 X 1 > 1, pi i=1 1

and nally a large positive integer n2 > n1 such that n1 ∞ X X 1 1 − > 1. i i p p i=n +1 1 i=1 1 2

Regardless of the rest of the sequence (nk ) this already ensures that f (p1 ) > 1.

Multiple common expansions in non-integer bases

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Second step. We choose p2 ∈ (1, p1 ) suciently small, e.g., p2 < 3/2, such that n ∞ 1 X X 1 1 − < −2, pi pi i=1 2 i=n +1 2 2

then a large positive integer n3 > n2 such that n1 n3 X X 1 1 − < −2, i i p p i=n +1 2 i=1 2 2

and nally a large positive integer n4 > n3 such that n1 n3 ∞ X X X 1 1 1 − + < −2. i i i p p p i=1 2 i=n +1 2 i=n +1 2 2

4

Regardless of the rest of the sequence (nk ) this already ensures that f (p2 ) < −2. Third step. We choose p3 ∈ (1, p2 ) suciently small, e.g., p3 < 4/3, such that n n ∞ 1 3 X X X 1 1 1 − + > 3, i i p p pi i=1 3 i=n +1 3 i=n +1 3 2

4

then a large positive integer n5 > n4 such that n3 n5 n1 X X X 1 1 1 − + > 3, i i i p p p i=n +1 3 i=n +1 3 i=1 3 2

4

and nally a large positive integer n6 > n5 such that n1 n3 n5 ∞ X X X X 1 1 1 1 − + − > 3. i i i p p p pi i=1 3 i=n +1 3 i=n +1 3 i=n +1 3 2

4

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This already ensures that f (p3 ) > 3. Fourth step. We choose p4 ∈ (1, p3 ) suciently small, e.g., p4 < 5/4, such that n n n ∞ 3 5 1 X X X 1 X 1 1 1 − + − < −4, i i i p p p pi i=1 4 i=n +1 4 i=n +1 4 i=n +1 4 2

4

6

then a large positive integer n7 > n6 such that n3 n5 n7 n1 X X X X 1 1 1 1 − + − < −4, i i i p p p pi i=n +1 4 i=n +1 4 i=n +1 4 i=1 4 2

4

6

and nally a large positive integer n8 > n7 such that n1 n3 n5 n7 ∞ X X X X X 1 1 1 1 1 − + − + < −4. i i i i p p p p pi i=n +1 4 i=1 4 i=n +1 4 i=n +1 4 i=n +1 4 2

4

6

8

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This already ensures that f (p4 ) < −4. Continuing this construction we end up with a continuous function satisfying f (pk ) > k for k = 1, 3, 5, . . . and f (pk ) < −k for k = 2, 4, 6, . . . . Since pk → 1 by construction, the required limit relations follow. Finally we observe that during the construction of the sequence (nk ) we had in each step more than one choice (in fact, innitely many choices). Hence there is a continuum of such sequences.

4 Concluding remarks and open questions The proof of Theorem 3 is based on appropriate properties of the polynomial f , namely that its coecients belong to the set {−1, 0, 1} and that it has many real roots with odd multiplicity. The Fekete polynomials Fp (t) =

p−1   X j j=1

p

tj

 

where p is an odd prime and pj denotes the Legendre symbol, are well known examples having the rst two properties. Baker and Montgomery [1], see also [2], proved that Fp (t) can have arbitrary many real roots, provided p is large enough. Dividing Fp (t) by t we get a family of polynomials with the rst two properties. Unfortunately we have no information about the multiplicity of their real roots. However the construction of the polynomial f can be generalized. Indeed, assume that h(t) ∈ {−1, 0, 1}[t] of degree d that has u ≥ 1 simple real roots in Qr k (0, 1). Let r ≥ 1 and set f (t) = k=1 h(t(d+1) ). Then f (t) ∈ {−1, 0, 1}[t] and it has at least ru simple real roots in (0, 1). By Exercise 46. of Section 5 of [6] the polynomial F163 (t) has two simple real roots in (0, 1). A straightforward computation with MAPLE shows that the same holds forF43 (t), too, and that F547 (t) has four simple real roots in the same interval. Hence these polynomial can replace 1 − t − t2 in the construction of f (t). Concerning Theorem 4 the following natural questions can be asked: (i) The theorem and its proof can be easily extended to all alphabets containg the zero digit, and both positive and negative digits. Does it remain true for two-letter alphabets {a, b} with a < 0 and b > 0? (ii) How slowly may converge a sequence (qj ) to one? (iii) Can we choose x and at least one base q1 ∈ (1, 1 + |x|) arbitrarily in the theorem? (iv) How large is the set of sequences (ci ) in {−1, 0, 1}∞ for which the corresponding function f satises the relations (6)?

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References 1. R.C. Baker and H.L. Montgomery, Oscillations of quadratic L-functions. Analytic number theory (Allerton Park, IL, 1989), 2340, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. 2. B. Conrey, A. Granville, B. Poonen and K. Soundararajan, Zeros of Fekete polynomials, Ann. Inst. Fourier (Grenoble) 50 (2000), 865889. 3. S. Kakeya, On the set of partial sums of an innite series, Proc. Tokyo Math.-Phys. Soc (2) 7 (1914), 250251. 4. S. Kakeya, On the partial sums of an innite series, Tôhoku Sc. Rep. 3 (1915), 159163. 5. V. Komornik and A. Peth®, Common expansions in non-integer bases, Publ. Math. Debrecen, 85 (2014), 489501. 6. G. Pólya and G. Szeg®, Problems and Exercises in Analysis, Vol. I, II Springer-Verlag, Berlin, New York, 1972. 7. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Hungar. 8 (1957), 477493.