Akiyama, Shigeki

Brunotte, Horst

Pethő, Attila

Steiner, Wolfgang

Thuswaldner, Jörg M.

University of Debrecen Electronic Archive

Open Problems in Mathematics, Volume 2 (2014)
PROBLEMS AND CONJECTURES AROUND SHIFT RADIX SYSTEMS
SHIGEKI AKIYAMA, HORST BRUNOTTE, ATTILA PETHO˝, WOLFGANG STEINER, AND JO¨RG M. THUSWALDNER
Abstract. Some basic open problems and conjectures concerning shift radix systems are listed and their
relations to well-known concepts and questions are outlined.
Original proposers of the open problem: Shigeki Akiyama, Horst Brunotte, Attila Petho˝, Wolfgang
Steiner and Jo¨rg M. Thuswaldner.
The year when the open problem was proposed : 2014.
Sponsor of the submission: Arturas Dubickas (Vilnius University).
1. Introduction and basic definitions
In 2005 Akiyama et al. [2] introduced the notion of a shift radix system and pointed out connections of
this simple dynamical system to well-known number systems such as beta-numeration and canonical number
systems. Let us first recall the definitions (here for y ∈ R we denote by byc the largest n ∈ Z with n ≤ y).
Definition 1.1. Let d ∈ N and r = (r0, . . . , rd−1) ∈ Rd.
(i) The mapping τr : Zd → Zd given by
τr(z) = (z1, . . . , zd−1,−brzc)t (z = (z0, . . . , zd−1)t ∈ Zd)
is called a shift radix system (SRS for short), where we set rz := r0z0 + · · ·+ rd−1zd−1.
(ii) We say that τr has the finiteness property if for each z ∈ Zd there is k ∈ N such that the k-fold
iterate of the application of τr to z satisfies τ
k
r (z) = 0.
This definition agrees with the one in [11], but the SRS in [2] coincide with our SRS with finiteness
property. Our definition is equivalent to the property that τr(z0, . . . , zd−1) = (z1, . . . , zd−1, zd)t, where zd is
the unique integral solution of the linear inequality
0 ≤ r0z0 + · · ·+ rd−1zd−1 + zd < 1 .
Therefore, we can write the mapping τr as the sum of a linear function and a small error term. More
explicitly, we have
τr(z) = Rr z + v(z) (z ∈ Zd),
where we put v(z) := (0, . . . , 0, rz − brzc)t (in particular, ||v(z)||∞ < 1) and Rr is a companion matrix of
the polynomial
χr(X) := X
d + rd−1Xd−1 + · · ·+ r1X + r0 (r = (r0, . . . , rd−1) ∈ Rd).
2010 Mathematics Subject Classification. 37.
The third author was partially supported by the OTKA grants No. 100339 and 104208.
1
2 S. AKIYAMA, H. BRUNOTTE, A. PETHO˝, W. STEINER, AND J. M. THUSWALDNER
Since χr agrees with the characteristic polynomial of the linear recurrence
zn = −rd−1zn−1 − · · · − r0zn−d,
SRS is viewed as an almost linear recurrence.
SRS and their relation to beta-numeration seem to have appeared for the first time in Hollander’s PhD
thesis [15] in 1996; two years earlier Vivaldi [26] had studied similar mappings in his detailed investigation
of discretized rotations (see also Reeve-Black and Vivaldi [22]). The dynamical aspects of SRS are described
in a broader context by Barat et al. [9].
The aim of the present paper is to provide a concise list of open problems and conjectures concerning
SRS thereby extending work of the first [1] and the third authors [21, Section 6]. The reader is referred to
[17] for detailed background information, illustrations and algorithms.
2. Problems and conjectures
The following sets play a central role in the study of various aspects of SRS.
Definition 2.1. For d ∈ N set
Dd :=
{
r ∈ Rd : ∀z ∈ Zd ∃k ∈ N ∃` ∈ N>0 : τkr (z) = τk+`r (z)
}
and
D(0)d :=
{
r ∈ Rd : τr is an SRS with finiteness property
}
.
Our fundamental open problem can roughly be described in the following way:
Problem 2.2. Give a complete description of Dd and D(0)d for each d ≥ 2.
In the sequel we break up Problem 2.2 into several subproblems and conjectures. Computational experi-
ments suggest the following (cf. [21]):
Problem 2.3. Prove that r ∈ D(0)d ∩Qd cannot be verified in polynomial time. Is it true that this problem
does not belong to the NP complexity class?
SRS are closely related to number systems:
• For an algebraic integer β > 1 the restriction of the beta-transformation Tβ to Z[β] is conjugate to
an SRS associated to a parameter defined by β.
• Akiyama et al. [5] investigated number systems with rational bases and established relations of these
number systems to Mahler’s 32 -problem (cf. [20]). These number systems can also be regarded as
special cases of SRS (see Steiner and Thuswaldner [25]) and there seem to be relations between the
3
2 -problem and the length of SRS tiles (see below) associated with τ−2/3.
• The backward division mapping used to define canonical number systems1 is conjugate to τr for
certain parameters r; thus, characterizing all bases of canonical number systems is a special case
of describing certain vectors r ∈ Qd giving rise to SRS with finiteness property (cf. Akiyama et
al. [2]).
1In [16] the term “complete base” was coined.
PROBLEMS AND CONJECTURES AROUND SHIFT RADIX SYSTEMS 3
Problem 2.4. Characterize all parameters r for which the digits of the underlying number systems gives a
language of a sofic shift (see [18] for the definition).
The Schur-Cohn region Ed (see Schur [24]) is the set of all vectors r ∈ Rd which define a contractive
polynomial χr.
Conjecture 2.5. We have D(0)d ⊂ Ed.
This conjecture has only been settled for d ∈ {1, 2, 3} (see [4, 14]).
Problem 2.6. What is the measure of D(0)d ? What can we say about the topology of D(0)d ? What is the
Hausdorff dimension of the boundary of D(0)d ?
These problems have only been settled for d = 1.
Results on the topology of D(0)2 are given by Weitzer [27], in particular, he showed that D(0)2 is neither
connected nor simply connected, and explicitly exhibited “holes” and components. It would be interesting
to prove the following conjecture.
Conjecture 2.7. The fundamental group of D(0)3 is non-trivial, i.e., it has a handle.
The following task was put forward and commented in [21].
Problem 2.8. Given r ∈ Ed ∩Qd, is r ∈ Dd algorithmically decidable?
Periodicity of the orbits on ∂Dd is an open problem even in dimension 2. For instance, the special case
(1, λ) ∈ ∂D2 can be expressed in the following simple arithmetical form:
Conjecture 2.9. Let λ ∈ R be such that −2 < λ < 2. Further, let a0, a1 ∈ Z and define
an+1 := −an−1 − bλanc (n ≥ 1).
Then the sequence a0, a1, a2, . . . is periodic.
Some partial results have been obtained in [3] (see also [19, 6]). A weaker conjecture is that for any λ
there exist infinitely many non-symmetric periodic orbits.
A remarkable conjecture of Schmidt [23] can be reformulated more generally in terms of the ultimate
periodicity of τr, where r belongs to the hypersurface
E
(C)
d := {r ∈ ∂Ed : Rr has a non-real eigenvalue of modulus 1}.
Problem 2.10. For which r ∈ E(C)d is each orbit of τr ultimately periodic, in particular the orbit of
(1, 0, . . . , 0)t ?
For particular orbits addressed in Problem 2.10 we suspect the following more explicit result.
Conjecture 2.11. Let r ∈ E(C)d be such that χr is irreducible. Let s be the number of pairs of complex
conjugate roots (α, α¯) of χr with |α| = 1. Then every orbit of (1, 0, . . . , 0)t under τr is periodic if s ∈ {1, 2},
and there exist r such that the orbit of (1, 0, . . . , 0)t is aperiodic if s ≥ 3 (see also [12], [13]).
4 S. AKIYAMA, H. BRUNOTTE, A. PETHO˝, W. STEINER, AND J. M. THUSWALDNER
SRS also admit a geometric theory, in particular, it is possible to define so-called SRS tiles (see Berthe´
et al. [11]). In the sequel we assume r0 6= 0.
Definition 2.12. Let r = (r0, . . . , rd−1) ∈ Ed and z ∈ Zd be given. The set
Tr(z) = Lim
n→∞ R
n
r τ
−n
r ({z})
is called the SRS tile associated with r; here the limit is taken with respect to the Hausdorff metric.
A tiling is a collection of compact sets covering Rd with zero measure overlaps (see [25]). SRS tiles are
conjectured to induce tilings of their representation spaces. A special case of this conjecture implies the
Pisot substitution conjectures (see e.g. Arnoux and Ito [7], Baker-Barge-Kwapisz [8], Barge [10]) for Pisot
beta substitutions. We present two challenging tasks.
Conjecture 2.13. Let r ∈ Ed. Then {Tr(z) : z ∈ Zd} is a tiling of Rd.
Problem 2.14. Let r ∈ Ed and z ∈ Zd. Give criteria for Tr(z) being the closure of its interior.
Finally, we state a problem related to the connectivity of the so-called central SRS tiles Tr(0).
Problem 2.15. Describe the Mandelbrot sets
{r ∈ Ed : Tr(0) is connected}
for d ≥ 2.
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PROBLEMS AND CONJECTURES AROUND SHIFT RADIX SYSTEMS 5
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Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, JAPAN (zip:305-8571)
E-mail address: akiyama@math.tsukuba.ac.jp
Haus-Endt-Straße 88, D-40593 Du¨sseldorf, GERMANY
E-mail address: brunoth@web.de
Faculty of Informatics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, HUNGARY
E-mail address: pethoe@inf.unideb.hu
LIAFA, CNRS UMR 7089, Universite´ Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, FRANCE
E-mail address: steiner@liafa.jussieu.fr
Chair of Mathematics and Statistics, University of Leoben, A-8700 Leoben, AUSTRIA
E-mail address: joerg.thuswaldner@unileoben.ac.at


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