Table Of ContentREGULARITIES OF THE DISTRIBUTION OF ABSTRACT VAN DER
CORPUT SEQUENCES
WOLFGANG STEINER
0
1
Abstract. Similarly to β-adic van der Corput sequences, abstract van der Corput se-
0
quences can be defined by abstract numeration systems. Under some assumptions, these
2
sequencesarelowdiscrepancysequences. Thediscrepancyfunctioniscomputedexplicitly,
n
and the bounded remainder sets of the form [0,y) are characterized.
a
J
3
2
1. Introduction
]
T
Let (x ) be a sequence with x ∈ [0,1) for all n ≥ 0, and
n n≥0 n
N
D(N,I) = #{0 ≤ n < N : x ∈ I}−Nλ(I)
. n
h
t its discrepancy function (or local discrepancy) on the interval I, where λ(I) is the length
a
m of I. Then, (x ) is said to be a low discrepancy sequence if sup D(N,I) = O(logN),
n n≥0 I
[ where the supremum is taken over all intervals I ⊆ [0,1). If D(N,I) is bounded in N, then
I is called a bounded remainder set. For details on discrepancy, we refer to [KN74, DT97].
2
v References toresults onboundedremainder setscanbefoundintheintroductionof[Ste06].
4
In [BG96, Nin98a, Nin98b], β-adic van der Corput sequences are defined, and it is
9
9 shown that they are low discrepancy sequences if β is a Pisot number with irreducible
3 β-polynomial. Recall that a Pisot number is an algebraic integer greater than 1 with all its
.
9 conjugates lying in the interior of the unit disk. We refer to Section 3 for the definition of
0
the β-polynomial. In [Mor98, IM04], these results were extended to piecewise linear maps
8
0 which generalize the β-transformation. The proof in [Nin98a, Nin98b] relies on the fact
: thatcylinder setsoftheβ-transformationareboundedremainder setsifβ isaPisot number
v
i with irreducible β-polynomial. Under the same conditions on β, bounded remainder sets
X
of the form [0,y), 0 ≤ y ≤ 1, were completely characterized in [Ste06]: the β-expansion
r
a of y is finite or its tail is the same as that of the expansion of 1.
If β is a Pisot number, then the language of the β-expansions is regular, which means
that it is recognized by a finite automaton. Therefore, these β-expansions are special cases
of abstract numeration systems as defined in [LR01, LR02], see Section 3. In Section 2,
we define van der Corput sequences related to more general abstract numeration systems.
Theorem 1 in Section 4 provides a new class of low discrepancy sequences. Finally, Theo-
rem 2 in Section 5 characterizes the bounded remainder sets of the form [0,y) with respect
to these abstract van der Corput sequences, generalizing the results in [Ste06].
Date: January 25, 2010.
2000 Mathematics Subject Classification. 11K38,11K31,11K16, 37B10,68Q45.
Thisworkwassupportedbythe FrenchAgence Nationalede laRecherche,grantANR–06–JCJC–0073.
1
2 WOLFGANGSTEINER
2. Definitions and first results
Let (A,≤) be a finite and totally ordered alphabet. Denote by A∗ the free monoid
generated by A for the concatenation product, i.e., the set of finite words with letters in A.
The length of a word w ∈ A∗ is denoted by |w|. Extend the order on A to A∗ by the
shortlex (or genealogical) order, that is to say v ≤ w if v = w or v < w, where v < w
means that either |v| < |w| or |v| = |w| and there exist p,v′,w′ ∈ A∗, a,b ∈ A such that
v = pav′, w = pbw′ and a < b.
According to [LR01], the triple S = (L,A,≤) is an abstract numeration system if L is
an infinite regular language over A, and the numerical value of a word w ∈ L is defined by
val (w) = #{v ∈ L : v < w}.
S
If val (w) = n, then we say that w is the representation of n and write rep (n) = w.
S S
Denote by Aω the set of (right) infinite words with letters in A. It is ordered by the
lexicographical order, that is to say t ≤ u if t = u or t < u, where t < u means that there
exist p ∈ A∗, a,b ∈ A, t′,u′ ∈ Aω, such that t = pat′, u = pbu′ and a < b.
Assume that the language L grows exponentially, with
log#{v ∈ L : |v| ≤ k}
lim = logβ > 0.
k→∞ k
Suppose that u ∈ Aω is the limit of words w(k) ∈ L, i.e., every finite prefix of u is a prefix
of w(k) for all but a finite number of k’s. Then, the value of u is the real number
val (w(k))
(1) valω(u) = lim S ,
S k→∞ #{v ∈ L : |v| ≤ |w(k)|}
if this limit exists and does not depend on the choice of w(k). Conditions assuring the
existence of this value are given in [LR02], see also Lemma 3 and its proof. Let
L = u ∈ Aω : u = lim w(k) for some w(k) ∈ L .
ω
k→∞
Since valω(u) ∈ [1/β,1], w(cid:8)e define the normalized value (cid:9)
S
βvalω(u)−1
hui = S ∈ [0,1].
β −1
We extend this definition to finite words w ∈ L which are prefixes of words in L by setting
ω
hwi = hui, where u is the smallest word in L with prefix w. Since we want to define a
ω
sequence without multiple occurrences of the same value, we set
L′ = {w ∈ L : hwi =6 hvi for every v ∈ L with v < w}.
Recall that the mirror image of a word w = w w ···w , w ∈ A, is w = w ···w w
1 2 k j k 2 1
and that the mirror image of a language L is L = {w : w ∈ L}. Now, we are ready to
define the main object of this paper, abstract van der Corput sequences. e
Definition 1 (Abstract van der Corput sequeence).eLet S = (L,A,≤) be an abstract
numeration system, where L is a regular language of exponential growth, every word w ∈ L
REGULARITIES OF THE DISTRIBUTION OF ABSTRACT VAN DER CORPUT SEQUENCES 3
is the prefix of some infinite word u ∈ L , and the limit in (1) exists for every u ∈ L .
ω ω
Then, the abstract van der Corput sequence corresponding to S is given by
xn = hwi with w = repSe′(n),
where S′ is the abstract numeration system L′,A,≤ .
e
Thus, the set of values of an abstract va(cid:0)n der Co(cid:1)rput sequence is {xn : n ≥ 0} =
e e
{hwi : w ∈ L} = {hwi : w ∈ L′}, and the position of hwi, w ∈ L′, in the sequence is
determined by the shortlex order on the mirror image of L′. We need a number of further
assumptions on the language L in order to get precise formulae for the discrepancy. All
these assumptions are satisfied by the β-adic van der Corput sequence when the language
of the β-expansions is regular, cf. Section 3, and by Example 1 at the end of this section.
Let A = (Q,A,τ,q ,F) be a (complete) deterministic finite automaton recognizing L,
L 0
with set of states Q, transition function τ : Q × A → Q, initial state q and set of final
0
states F. The transition function is extended to words, τ : Q × A∗ → Q, by setting
τ(q,ε) = q for the empty word ε and τ(q,wa) = τ(τ(q,w),a). A word w ∈ A∗ is accepted
by A , and thus in L, if and only if τ(q ,w) ∈ F.
L 0
Definition 2 (Totally ordered automaton). A deterministic automaton (Q,A,τ,q ,F) is
0
said to be a totally ordered automaton if there exists a total order on the set of states Q
such that, for all q,r ∈ Q,
q ≤ r implies τ(q,a) ≤ τ(r,a) for every a ∈ A.
From now on, all automata will be totally ordered automata. Furthermore, the maximal
state will be the initial state and every state except the minimal one will be final. (In case
Q = F, where the automaton recognizes A∗, we add a non-accessible state to Q.) W.l.o.g.,
the set of states will be Q = {0,1,...,d} for some positive integer d, and the order on Q
will be the usual order on the integers, hence q = d and F = {1,...,d}. Moreover, we
0
will assume that τ(0,a) = 0 for every a ∈ A, i.e., the state 0 is a sink.
Lemma 1. Let L ⊆ A∗ be recognized by a totally ordered automaton A = (Q,A,τ,d,Q\
L
{0}), with Q = {0,1,...,d} and τ(0,a) = 0 for every a ∈ A. Then, L is recognized by the
totally ordered automaton Ae = (Q,A,τ,0,Q\{0}), where
L
e
(2) τ(r,a) = # q ∈ Q : τ(q,a)+r > d for every r ∈ Q, a ∈ A.
e
In particular, we have τ(0,a) = 0 for every a ∈ A.
(cid:8) (cid:9)
e
Proof. A deterministic automaton A′ = (Q′,A,τ′,q′,F′) recognizing L is obtained by
0
e
determinizing the automaton which is given by inverting the transition function τ, see e.g.
[Sak09]. This means that e
• Q′ is a subset of the power set P(Q),
• τ′(q′,a) = {q ∈ Q : τ(q,a) ∈ q′} for every q′ ∈ Q′, a ∈ A,
• the set of final states in A is the initial state of A′, i.e., q′ = {1,...,d},
L 0
• the final states in A′ are those elements of Q′ which contain the initial state of A ,
L
i.e., F′ = {q′ ∈ Q′ : d ∈ q′}.
4 WOLFGANGSTEINER
We show that Q′ ⊆ {q′,q′,...,q′}, where q′ = {r + 1,...,d} (q′ being the empty set).
0 1 d r d
Since A is totally ordered and τ(0,a) = 0, we obtain that
L
τ′(q′,a) = {q ∈ Q : τ(q,a) > 0} = {r +1,...,d} = q′ for some r ∈ Q.
0 r
In the same way, we get, for every r ∈ Q with q′ ∈ Q′, that
r
τ′(q′,a) = {q ∈ Q : τ(q,a) > r} = {s+1,...,d} = q′ for some s ∈ Q.
r s
This shows that Q′ ⊆ {q′,q′,...,q′}. It is easy to see that A′ is a totally ordered au-
0 1 d
tomaton, with the order on Q′ given by q′ ≤ q′ if q′ ⊆ q′, i.e., r ≥ s. We clearly have
r s r s
F′ = {q′,q′,...,q′ }∩Q′. If we extend the set of states to {q′,q′,...,q′} (with possibly
0 1 d−1 0 1 d
non-accessible states) and label the states by d−r instead of qr′, we obtain ALe. Therefore,
Ae is a totally ordered automaton, with Q ordered by the usual order on the integers. (cid:3)
L
The next lemma provides a fundamental characterization of the words in a language L
recognized by a totally ordered automatonA = (Q,A,τ,d,Q\{0}) with Q = {0,1,...,d}
L
and τ(0,a) = 0 for every a ∈ A.
Lemma 2. Let L,τ,τ be as in Lemma 1, w ···w ∈ A∗, 0 ≤ j ≤ k. We have w ···w ∈ L
1 k 1 k
if and only if τ(d,w ···w )+τ(d,w ···w ) > d.
1 j k j+1
e
Proof. Let 0 ≤ j ≤ k. With the notation of the proof of Lemma 1, τ(d,w ···w ) =
k j+1
d− r can be written as τ′(q′,we ···w ) = q′ = {r + 1,...,d}. In A , this means that
0 k j+1 r L
w ···w leads to a final state from the state q, i.e., τ(q,w ···w ) > 0, if and only if
j+1 k j+1 k e
q > r. Therefore, we have τ(d,w ···w ) = τ(τ(d,w ···w ),w ···w ) > 0 if and only if
1 k 1 j j+1 k
τ(d,w ···w ) > r, i.e., τ(d,w ···w )+τ(d,w ···w ) > d. (cid:3)
1 j 1 j k j+1
Remark 1. If we consider τ(d,a)+···+τ(1,a) as a partition of an integer, then τ(d,a)+
e
··· + τ(1,a) is the conjugate partition, since τ(d − r,a) = # q ∈ Q : τ(q,a) > r .
E.g., if (τ(d,a),...,τ(1,a)) = (4,2,1,0), then (τ(d,a),...,τ(1,a)) = (3,2,1,1), and the
e
(cid:8) (cid:9)
corresponding Ferrers diagram is
e e
3 2 1e1 e
4
2 .
1
0
Next, we characterize the values of the abstract van der Corput sequence, under the
assumption that the incidence matrix of the co-accessible part of A is primitive. (A state
L
q is co-accessible if τ(q,w) ∈ F for some w ∈ A∗.)
Lemma 3. Let L be as in Lemma 1 and assume that M = (#{a ∈ A : τ(q,a) = r})
L 1≤q,r≤d
is a primitive matrix. Then, the normalized value exists for every u ∈ L and is given by
ω
∞
(3) hui = ǫ (u)β−j with ǫ (u) = η ,
j j τ(d,u1···uj−1a)
Xj=1 aX<uj
REGULARITIES OF THE DISTRIBUTION OF ABSTRACT VAN DER CORPUT SEQUENCES 5
where β is the Perron-Frobenius eigenvalue of M , (η ,...,η )t is the corresponding right
L 1 d
(column) eigenvector with η = 1, η = 0, and u = u u ··· with u ∈ A for all j ≥ 1.
d 0 1 2 j
Proof. The definition and primitivity of the incidence matrix M give
L
(4) #{v ∈ Ak : τ(q,v) > 0} = (0,...,1,...,0)Mk(1,...,1)t = cη βk +O(ρk)
L q
with constants c > 0 and ρ < β such that every eigenvalue α 6= β of M satisfies |α| < ρ.
L
Due to the assumptions on A , we have u ···u ∈ L for every k ≥ 1. Similarly to
L 1 k
[LR01, LR02], we split up
{v ∈ L : v < u ···u } = {v ∈ L : |v| < k}∪ {u ···u aw ∈ L : w ∈ Ak−j}.
1 k 1 j−1
1≤[j≤ka[<uj
Since #{v ∈ L : |v| ≤ k} = k (cβj +O(ρj)) = cβk+1 +O(max(1,ρ)k), we obtain
j=0 β−1
val (u ···u ) P1 β −1 k max(1,ρ)k
S 1 k = + η β−j +O .
#{v ∈ L : |v| ≤ k} β β τ(d,u1···uj−1a) βk
Xj=1 aX<uj (cid:18) (cid:19)
Therefore, we have
∞
val (u ···u ) 1 β −1
valω(u) = lim S 1 k = + ǫ (u)β−j,
S k→∞ #{v ∈ L : |v| ≤ k} β β j
j=1
X
and hui = (βvalω(u)−1)/(β −1) yields (3). (cid:3)
S
As a last preparation for the study of the discrepancy of abstract van der Corput se-
quences, we consider the language L′.
Lemma 4. Let L be as in Lemma 3 and assume that τ(q,a ) > 0 for every q > 0, where
0
a denotes the smallest letter of A. Then, L′ consists exactly of those words in L which do
0
not end with a .
0
Proof. We clearly have hti ≤ hui if t < u, t,u ∈ L . The primitivity of M implies that
ω L
η > 0 for all q > 0. Therefore, hti = hui if and only if there exists no word u′ ∈ L with
q ω
t < u′ < u. For v < w, v,w ∈ L, this means that hvi = hwi if and only if v is a prefix of w
and w is the smallest right extension of v in L of length |w|. Since τ(q,a ) > 0 for every
0
q > 0, we have vak ∈ L for all k ≥ 0. It follows that hvi = hwi with v < w, v,w ∈ L, if
0
and only if w = va|w|−|v|. Thus, w 6∈ L′ if and only if w ends with a . (cid:3)
0 0
Example 1. Let A = ({0,1,2,3},{a ,a ,a },τ,3,{1,2,3})be the totally ordered automa-
L 0 1 2
ton in Figure 1 on the left. The first words in L (in the shortlex order) are
ε,a ,a ,a ,a a ,a a ,a a ,a a ,a a ,a a ,a a ,a a a ,a a a ,a a a ,a a a ,a a a ,
0 1 2 0 0 0 1 0 2 1 0 1 2 2 0 2 2 0 0 0 0 0 1 0 0 2 0 1 0 0 1 2
a a a ,a a a ,a a a ,a a a ,a a a ,a a a ,a a a ,a a a ,a a a ,a a a ,a a a .
0 2 0 0 2 2 1 0 0 1 0 1 1 0 2 1 2 0 1 2 2 2 0 0 2 0 2 2 2 0 2 2 2
6 WOLFGANGSTEINER
a0 a0,a2
a0
a0
3 2 3 2
a1
a2 a1
a2 a1 a1 a2
a0 a0
1 0 1 0
a1 a1,a2
a2 a0,a1,a2 a0,a1,a2
Figure 1. The totally ordered automata AL (left) and ALe (right) of Example 1.
The transition functions τ, τ and the incidence matrices ML, MLe are given by
τ a a a τ a a a
0 1 2 0 1 2
e
0 0 0 0 1 1 0 0 0 0 0 0 1 0
1 2 0 1 , ML = 1 0 1, e1 2 0 0 , MLe = 1 0 1.
2 3 0 1 1 1 1 2 3 1 0 1 0 2
3 3 2 1 3 3 1 3
Recall that ML = (#{a ∈ A : τ(q,a) = r})1≤q,r≤3, MLe = (#{a ∈ A : τ(q,a) = r})1≤q,r≤3
and that τ can be calculated using Remark 1. Thus, L is recognized by the totally ordered
automaton ALe in Figure 1 on the right. The characteristic polynomial oef ML is x3−2x2−
x+1, the dominant eigenvalue is β ≈ 2.247, and (η ,ηe,η )t = (β2−2β,−β2+3β−1,1)t ≈
e 1 2 3
(0.555,0.692,1)t is a right eigenvector of M . Since x3−2x2−x+1 is irreducible, it must
L
be the characteristic polynomial of Me as well. The conditions of Lemma 4 are satisfied,
L
hence the first elements of the abstract van der Corput sequence corresponding to L are
η η +η η
3 3 2 3
x = hεi = 0, x = ha i = , x = ha i = , x = ha a i = ,
0 1 1 β 2 2 β 3 0 1 β2
η +η η η η +η η
3 2 3 3 3 2 2
x = ha a i = , x = ha a i = + , x = ha a i = + ,
4 0 2 β2 5 1 2 β β2 6 2 2 β β2
η η η η +η
3 3 3 3 2
x = ha a a i = , x = ha a a i = + , x = ha a a i = ,
7 0 0 1 β3 8 1 0 1 β β3 9 0 0 2 β3
η η +η η +η η
3 3 2 3 2 3
x = ha a a i = + , x = ha a a i = + ,
10 1 0 2 β β3 11 2 0 2 β β3
η η η +η η
3 3 3 2 2
x = ha a a i = + , x = ha a a i = + ,
12 0 1 2 β2 β3 13 0 2 2 β2 β3
η η η η +η η η
3 3 2 3 2 2 2
x = ha a a i = + + , x = ha a a i = + + .
14 1 2 2 β β2 β3 15 2 2 2 β β2 β3
x0 x7 x9 x3 x12 x4 x13 x1 x8 x10 x5 x14 x2 x11 x6 x15
REGULARITIES OF THE DISTRIBUTION OF ABSTRACT VAN DER CORPUT SEQUENCES 7
3. β-adic van der Corput sequences
To obtain Ninomiya’s β-adic van der Corput sequences, consider a totally ordered au-
tomaton A = ({0,1,...,d},A,τ,d,{1,...,d}) on the alphabet A = {0,1,...,B}, with
L
integers b ∈ A, 1 ≤ q ≤ d, such that τ(q,a) = d for all a < b , and τ(q,a) = 0 if and only
q q
if a > b or q = 0. Assume that M is primitive (which is the only interesting case since
q L
τ(q,0) 6= d implies b = 0), and let β be its Perron-Frobenius eigenvalue. Then, we have
q
ǫ (u) = η = η = u for every u = u u ··· ∈ L , j ≥ 1.
j τ(q,u1···uj−1a) d j 1 2 ω
aX<uj aX<uj
Let t t ··· be the maximal sequence in L , i.e., t = b for all j ≥ 1. Since A
1 2 ω j τ(d,t1···tj−1) L
is a totally ordered automaton, τ(d,t ···t ) ≤ τ(d,t ···t ) implies t < t or t = t ,
1 j−1 1 k−1 j k j k
τ(d,t ···t ) ≤ τ(d,t ···t ), thus t t ··· ≤ t t ···, in particular t t ··· ≤ t t ···.
1 j 1 k j j+1 k k+1 j j+1 1 2
By the maximality of t t ···, we have ∞ t β−j = 1. The sequence t t ··· is called the
1 2 j=1 j 1 2
infinite expansion of 1 in base β or quasi-greedy expansion of 1, cf. [Par60]. By the special
P
structureofA andtheprimitivityofM ,wehave{τ(d,t ···t ) : 0 ≤ j < d} = {1,...,d},
L L 1 j
hence τ(d,t ···t ) = τ(d,t ···t ) for some m < d. Therefore, t t ··· is eventually
1 d 1 m 1 2
periodic with preperiod length m and period length d−m, which implies
d m
1− t β−j = 1− t β−j βm−d,
j j
j=1 (cid:18) j=1 (cid:19)
X X
thus β is a root of the polynomial
(5) (xd −t xd−1 −···−t x0)−(xm −t xm−1 −···−t x0).
1 d 1 m
If A is the minimal deterministic automaton recognizing L or, equivalently, m and d−m
L
are the minimal preperiod and period lengths of t t ···, then (5) is called β-polynomial.
1 2
If u = u u ··· ∈ L , then we have either u = t t ··· or u ···u = t ···t , u < t
1 2 ω 1 2 1 k−1 1 k−1 k k
for some k ≥ 1. Since τ(d,u ···u ) = d in the latter case, u ∈ L is equivalent with
1 k ω
u u ··· ≤ t t ··· for all j ≥ 1, cf. [Par60]. Therefore, u is either the greedy or the
j j+1 1 2
quasi-greedy β-expansion of hui. Since w00··· is a greedy β-expansion for every w ∈ L,
the abstract van der Corput sequence given by S = (L,A,≤) is exactly the β-adic van der
Corput sequence defined in [Nin98a]. Consequently, we call A a β-automaton.
L
Conversely, let t t ··· be the infinite expansion of 1 in base β > 1, i.e., the unique
1 2
sequence of integers satisfying ∞ t β−j = 1 and 00··· < t t ··· ≤ t t ··· for all
j=1 j j j+1 1 2
j ≥ 1, cf. [Par60]. Assume that t t ··· is eventually periodic. In particular, this holds
1 2
P
when β is a Pisot number, see [Ber77, Sch80]. Let d be the sum of the minimal preperiod
and period lengths, and q = #{1 ≤ k ≤ d : t t ··· ≤ t t ···} for j ≥ 0. Then, the
j k k+1 j+1 j+2
β-automaton is given by b = t and τ(q ,t ) = q , see Example 2.
qj j+1 j j+1 j+1
Note that β-adic van der Corput sequences were first considered in [BG96] for some
cases in which L = L. In this case, the definition of the sequence is simpler beacause the
β-expansion of x is the mirror image of the expansion of n in a numeration system with
n
respect to the lienear recurrence corresponding to the β-polynomial. In our notation, L = L
is equivalent with τ = τ, see also [BY00, Kwo09] for a different characterization.
e
e
8 WOLFGANGSTEINER
Example 2. Let β be the real root of x3 −4x2 −2. Then, t t ··· = 401401···, hence we
1 2
have d = 3, q = 3, q = 1, q = 2, q = 3, thus b = 4, b = 0, b = 1, and
0 1 2 3 3 1 2
τ 0 1 2 3 4 τ 0 1 2 3 4
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 2 0
1 2 0 0 0 0 , ML = 0 0 2, e1 2 2 1 1 0 , MLe = 2 1 1.
2 3 3 0 0 0 1 0 4 2 3 2 1 1 0 3 1 1
3 3 3 3 3 1 3 3 2 1 1 1
This example is also given in Section 2.3 in [Ste06], with different notation. The substitu-
tion τ in [Ste06] plays the role of the transition function τ in this paper.
4. Discrepancy functioen
For a given abstract van der Corput sequence (x ) and y ∈ [0,1], we study now the
n n≥0
behavior of the function D(N,[0,y)) = #{0 ≤ n < N : x < y}−Ny as N → ∞. Since
n
D(N,[y,z)) = D(N,[0,z))−D(N,[0,y)), this determines the discrepancy of (x ) .
n n≥0
If L satisfies the conditions of Lemma 3, then every y ∈ [0,1] is the numerical value of
some u ∈ L , see [LR02]. We have the following lemma, where aω = a a ···.
ω 0 0 0
Lemma 5. Let (x ) be an abstract van der Corput sequence with L as in Lemma 4.
n n≥0
For y = hui with u = u1u2··· ∈ Lω, N ≥ 0 with repSe′(N) = wℓ···w1, we have
ℓ ℓ
#{0 ≤ n < N : x < y} = #Lk−j−1 +C(N,u),
n τ(d,u1···uj−1a),τe(d,wℓ···wk+1b)
Xj=1 aX<ujkX=j+1bX<wk
where Lm = {v ∈ Am : τ(q,v)+r > d} = {v ∈ Am : q +τ(r,v) > d} and
q,r
ℓ
C(N,u) = #{b < w : bw ···w aω < u u ··· ,eu ·e··u bw ···w ∈ L}.
k k+1 ℓ 0 k k+1 1 k−1 k+1 ℓ
k=1
X
Since ℓ = O(logN), we have C(N,u) = O(logN).
Proof. We have to count the number of words v ∈ L′ with valSe′(v) < N, i.e., v < wℓ···w1,
ℓ−|v| ℓ−|v|
and hvi < y. As in the proof of Lemma 4, we have va ∈ L, thus hvi = hva i. Since
0 0
w > a , v < w ···w holds if and only if aℓ−|v|v < w ···w . Theerefore, we cean count the
ℓ 0 ℓ 1 0 ℓ 1
number of words v ∈ L∩Aℓ with v < w ···w and hvi < y, instead of those in L′.
ℓ 1
The inequality hvi < y is equivalent with vaω < u. Thus, we have to count the v in L
e 0e
of the form v = u ···u av ···v bw ···w with a < u , b < w , 1 ≤ j < k ≤ ℓ, or
1 j−1 j+1 ek−1 k+1 ℓ j k
v = u ···u bw ···w with b < w , bw ···w aω < u u ···, 1 ≤ k ≤ ℓ.
1 k−1 k+1 ℓ k k+1 ℓ 0 k k+1
Lemma 2 yields that u ···u av ···v bw ···w ∈ Lif andonly if v ···v ∈
1 j−1 j+1 k−1 k+1 ℓ j+1 k−1
Lk−j−1 , which provides the main part of the formula. The other words
τ(d,u1···uj−1a),τe(d,wℓ···wk+1b)
give C(N,u). We have ℓ = O(logN) since L grows exponentially and the same holds for
L′ by Lemma 4. Since A is finite, we obtain C(N,u) = O(logN). (cid:3)
REGULARITIES OF THE DISTRIBUTION OF ABSTRACT VAN DER CORPUT SEQUENCES 9
Similarly to [Nin98a, Nin98b, Ste06], we assume now that the characteristic polynomial
of M is irreducible. Let β ,...,β be the conjugates of the Perron-Frobenius eigenvalue
L 2 d
β = β. For any z ∈ Q(β), denote by z(i) ∈ Q(β ) the image of z by the isomorphism
1 i
mapping β to β . Similarly to (4), we have some constants θ ,...,θ ∈ Q(β) such that
i 1 d
d
#Lk = (0,...,1,...,0)Mk(0,...,0,1,...,1)t = η(i)θ(i)βk for 1 ≤ q,r ≤ d.
q,r L q r i
i=1
X
Note that (θ1,...,θd)t is a right eigenvector of MLe. Set θ0 = 0 and define
(6) γk(N) = θτe(d,wℓ···wk+1b) for N = valSe′(wℓ···w1), 1 ≤ k ≤ ℓ,
b<w
Xk
similarly to ǫ (u). Then,
j
d
#Lk−j−1 = ǫ(i)(u)γ(i)(N)βk−j−1,
τ(d,u1···uj−1a),τe(d,wℓ···wk+1b) j k i
aX<ujbX<wk Xi=1
ℓ ℓ d d ℓ
N = #Lk−1 = η(i)θ(i) βk−1 = γ(i)(N)βk−1.
d,τe(d,wℓ···wk+1b) d τe(d,wℓ···wk+1b) i k i
Xk=1bX<wk Xk=1bX<wkXi=1 Xi=1 Xk=1
For y = hui, we have thus
ℓ ℓ ∞
D(N,[0,y)) = #Lk−j−1 +C(N,u)−N ǫ (u)β−j
τ(d,u1···uj−1a),τe(d,wℓ···wk+1b) j
Xj=1 aX<ujkX=j+1bX<wk Xj=1
∞ d ℓ ℓ
= ǫ(i)(u)γ(i)(N)βk−j−1− γ(i)(N)βk−1ǫ (u)β−j +C(N,u)
j k i k i j
!
j=1 i=1 k=j+1 k=1
XX X X
∞ ℓ d
(7) = γ(i)(N)βk−1 ǫ(i)(u)β−j −ǫ (u)β−j
k i j i j
Xj=1 kX=j+1Xi=2 (cid:16) (cid:17)
min(j,ℓ) d
− γ(i)(N)βk−1ǫ (u)β−j +C(N,u).
k i j
!
k=1 i=1
X X
Since the series converges absolutely, we can change the order of summation, and get
ℓ k−1 d
D(N,[0,y)) = γ(i)(N)βk−1 ǫ(i)(u)β−j −ǫ (u)β−j
k i j i j
Xk=1 Xj=1 Xi=2 (cid:16) (cid:17)
∞ d
(8) − γ(i)(N)βk−1ǫ (u)β−j +C(N,u).
k i j
!
j=k i=1
XX
The following theorem states that D(N,[0,y)) = O(logN) if |β | < 1 for 2 ≤ i ≤ d, i.e.,
i
if β is a Pisot number. The conditions on M are subsumed in the following definition.
L
10 WOLFGANGSTEINER
Definition 3 (Pisot automaton). A deterministic automaton is said to be a Pisot automa-
ton if the incidence matrix of its restriction to the states which are both accessible and
co-accessible has one simple eigenvalue β > 1, and all other eigenvalues satisfy |α| < 1.
Theorem 1. Let S = (L,A,≤) be an abstract numeration system where L is recognized by
a totally ordered Pisot automaton A = ({0,1,...,d},A,τ,d,{1,...,d}) with τ(0,a) = 0
L
for every a ∈ A, τ(q,a ) > 0 for the minimal letter a ∈ A and every q > 0. Then, the
0 0
corresponding abstract van der Corput sequence is a low discrepancy sequence.
Proof. If A is a Pisot automaton, then the characteristic polynomial of M is irreducible,
L L
thus M is primitive. Therefore, the conditions of Lemma 5 are satisfied, and D(N,[0,y))
L
is given by (8). Since γ (N) and ǫ (u) take only finitely many different values and ℓ =
k j
O(logN), we obtain
ℓ
D(N,[0,y)) = O(1)+O(logN) = O(logN),
k=1
X
where the constants implied by the O-symbols do not depend on y. With D(N,[y,z)) =
D(N,[0,z))−D(N,[0,y)),we getsup D(N,I) = O(logN), andthetheoremisproved. (cid:3)
I
AsSection3shows, theβ-adicvanderCorputsequences definedbyPisotnumbersβ with
irreducible β-polynomial considered in [Nin98a, Nin98b] are special cases of the abstract
van der Corput sequences in Theorem 1. By Lemma 2, S = (L,A,≤) is another, usually
different, abstract numeration system satisfying the conditions of Theorem 1 whenever
τ(q,a0) > 0 for every q > 0, in particular when AL is aeβ-auteomaton. The abstract van
der Corput sequence considered in Example 1 is also a new low discrepancy sequence.
e
5. Bounded remainder sets
Under the conditions of Theorem 1, (7) gives
(9) D(N,[0,y))
∞ ℓ d min(j,ℓ)
= γ(i)(N)ǫ(i)(u)βk−j−1− γ (N)ǫ (u)βk−j−1 +C(N,u)+O(1)
k j i k j
!
j=1 k=j+1 i=2 k=1
X X X X
for y = hui. If there exists some m ≥ 0 such that u = a for all j > m, then ǫ (u) = 0
j 0 j
for all j > m and C(N,u) is bounded. It follows that the interval [0,hvi) = [0,hvaωi) is a
0
bounded remainder set for every finite word v ∈ L.
For any β-adic van der Corput sequence where β is a Pisot number with irreducible
β-polynomial, the bounded remainder sets [0,y) have been characterized in [Ste06] by the
factthatthetailoftheβ-expansionofy iseither0ω orasuffixoft t ···. Withournotation
1 2
and the β-automaton defined in Section 3, this means that ∞ u βm−j = η for some
j=m+1 j q
m ≥ 0, q ∈ Q. In the more general case, we have the following partial characterization.
P
Note that the proof is simpler than the one in [Ste06].
