Van der Corput Sequences and Linear
Permutations
Florian Pausinger 1,2
IST Austria
Klosterneuburg, Austria
Abstract
This extended abstract is concerned with the irregularities of distribution of onedimensional permuted van der Corput sequences that are generated from linear
permutations. We show how to obtain upper bounds for the discrepancy and diaphony of these sequences, by relating them to Kronecker sequences and applying
earlier results of Faure and Niederreiter.
Keywords: Discrepancy, permuted van der Corput sequence, continued fraction,
Zaremba’s conjecture.
1
Introduction
The development of the theory of uniform distribution modulo 1 goes back
to a famous paper of Weyl in 1916 [24]. In its classical setting, this theory is
concerned with the distribution of fractional parts of sequences of real numbers
in the unit interval [0, 1). In 1935, van der Corput [22,23] added a quantitative
aspect to the theory when he defined discrepancy and initiated the study of
1
This research is supported by the Graduate school of IST Austria (Institute of Science
and Technology Austria).
2
Email: florian.pausinger@ist.ac.at
irregularities of distributions of finite point sets and infinite sequences. The
following years witnessed a rapid development of the theory, culminating in
the famous results of Roth [20] and Schmidt [21], which state lower bounds for
the discrepancy of arbitrary point sets and sequences, as well as the Theorem
of Erdös-Turán [5], which provides upper bounds for the discrepancy of finite
point sets; see also [4].
In the last 30 years, the focus in uniform distribution theory changed to
more applied questions. New algorithms for the construction of sequences
with small discrepancy improved numerical integration algorithms based on
quasi-Monte Carlo methods; see [4,15].
Our work is concerned with a generalization of the classical one-dimensional
van der Corput sequences due to Faure [7]. These sequences are generated from
arbitrary permutations in fixed integer bases, and their discrepancy is of the
optimal order of magnitude. As we shall see, a clever choice of generating permutations can considerably improve the distribution properties of permuted
van der Corput sequences, and hence their value for applications.
In the following paragraphs, we focus on linear generating permutations
of the form σ(i) = m · i (mod p), for a prime base p and a multiplier 1 ≤
m ≤ p − 1, and give a full classification of their distribution properties. The
proofs of our results can be found in [18], as well as in a recent preprint with
Topuzoglu [19], in which more general families of permutations defined by
particular permutation polynomials are studied.
2
Discrepancy Theory
Let I = [0, 1) be the half open unit interval and let J ⊆ I. For an infinite
sequence X = (xi )i≥1 in I and for N ≥ 1, let #(J, N, X) denote the number of
indices i ≤ N for which xi ∈ J. Let R(J, N, X) := #(J, N, X) − l(J)N denote
the remainder or discrepancy function, in which we write l(J) for the length
of the interval. We call an infinite sequence X uniformly distributed if
#(J, N, X)
= l(J),
N →∞
N
lim
for every subinterval J of I.
The extreme discrepancy, DN (X), and the diaphony, FN (X), of the first
N points of X are
DN (X) = sup |R([α, β), N, X)|,
[α,β)⊂I
FN (X) =
2π
2
Z Z
2
|R([α, β[, N, X)| dαdβ
I
1/2
;
I
see [9,25]. Note that a sequence X is uniformly distributed if and only if
lim DN (X)/N = 0 and lim FN2 (X)/N = 0. Answering a question of Erdös
N →∞
N →∞
from [6], Schmidt proved that there exists a positive constant c1 such that
every infinite sequence in [0, 1) satisfies
lim DN (X) > c1 log N.
N →∞
This is also true for FN2 (X). Hence, we say X is a low discrepancy (diaphony)
sequence if there exists a positive constant c2 such that for all N , DN (X) and
FN2 (X) < c2 log N. We compare two low discrepancy sequences by computing
the asymptotic values:
FN2 (X)
DN (X)
, f (X) := lim sup
,
t(X) := lim sup
N →∞ log N
N →∞ log N
and consider sequences with smaller value to be more regularly distributed.
This is motivated by the well-known Theorem of Koksma-Hlawka, which is
used for the numerical integration of a function on the multi-dimensional unit
cube. It gives an explicit bound for the integration error in terms of a product
of the (Hardy-Krause) variation times the discrepancy of the point set at
which the function is evaluated. Writing f for the function and Var(f ) for its
variation, the one-dimensional version of the theorem is known as Koksma’s
Inequality [11]:
Z
N
1 X
f (xi ) − f (x)dx ≤ Var(f ) · DN (X).
N
I
i=1
3
Permuted van der Corput Sequences
P
j
Let ∞
j=0 aj (k)b be the b-adic representation of the integer k ≥ 1, with 1 ≤
n
k < b and aj (k) = 0 if j ≥ n and let Sb denote the set of all permutations of
{0, 1, . . . , b − 1}. We follow Faure [7] and define the permuted van der Corput
sequence, Sbσ , for a fixed base b ≥ 2, all k ≥ 1, and a permutation, σ ∈ Sb , by
Sbσ (k) =
∞
X
σ(aj (k))
j=0
bj+1
.
(1)
Generalized Halton sequences are then multi-dimensional sequences, whose
i-th coordinate is determined by a permuted van der Corput sequence. Halton sequences are uniformly distributed if the bases of the generating van der
Corput sequences are coprime. The van der Corput sequence is also the prototype of other multi-dimensional sequences like digital (t, s)-sequences over
Zp as introduced by Niederreiter [14].
The sequences that are generated from identity permutations in arbitrary
bases are usually referred to as original van der Corput sequences even if van
der Corput only considered the sequences in base 2. It is remarkable that Faure
developed powerful methods to compute the asymptotic constants t(Sbσ ) and
f (Sbσ ) of arbitrary van der Corput sequences exactly. These computations,
although theoretically possible, can be tedious in practice; see [3,8,16,17].
Nevertheless, Chaix and Faure computed the asymptotic discrepancy and diaphony values for the original van der Corput sequences explicitely; see [7,
Théorème 6], [3, Théorème 4.13]. We simplify these results to
t(Sbid ) = c3 ·
b
,
log b
f (Sbid ) = c4 ·
b2
.
log b
(2)
In [10], Faure proved that the original van der Corput sequences always have
the worst distribution behavior in the family of all permuted van der Corput
sequences in a certain base b, which implies that the values in (2) are upper
bounds for any sequence in base b. Note that t(Sbid ) and f (Sbid ) depend on the
base b and increase with increasing base. In contrast, Faure also presented an
algorithm that generates a permutation σ in every integer base b such that
t(Sbσ ) < 1/ log 2; see [8]. These observations and results finally motivate our
main research questions:
1. Can we explicitely compute the asymptotic values for entire families of
permutations other than the identity?
2. Can we find explicit families of permutations in arbitrary bases, for which
the discrepancy constants of the corresponding sequences are independent
of the base?
We present answers to both questions in the next paragraph.
4
Results
Concerning the first question, we refer to our recent paper [18], which studies
the family of linear permutations of the form σ(i) = m·i (mod p), for a prime p
and a multiplier 1 ≤ m ≤ p − 1, as well as related permutations with a similar
structure. We compute explicit lower bounds for the asymptotic diaphony
constants of the set of linear permutations whose multipliers m divide either
p + 1 or p − 1, and we observe that these constants increase with the base p
like in the case of the identity permutation.
Theorem 4.1 For a prime base p, let 1 ≤ m ≤ p − 1 be such that p =
m · a + m − 1 or p = m · a + 1 for an integer a > 0. Then, for σ(i) = m · i
(mod p),
m 2 + a2
≤ f (Spσ ),
(3)
c5 ·
log p
in which the constant c5 depends on whether m is odd or even and whether a
is odd or even.
We conclude that multipliers m that either divide p + 1 or p − 1 generate
van der Corput sequences with weak distribution properties. In fact, these
sequences behave asymptotically similar to sequences generated from identity
permutations in smaller bases.
In order to approach our second question, we put Theorem 4.1 in a more
general context. Let ā = [0; a1 , a2 , . . . , an ] = [a1 , a2 , . . . , an ] be the continued
fraction expansion of a rational ā ∈ [0, 1], in which the ai are positive integers and called partial quotients. Since the continued fraction expansion of a
rational number is finite, there always exists a constant amax that bounds the
partial quotients from above.
Letting α = [0; α1 , α2 , . . .] be an irrational in [0, 1), we call the rational
[0; α1 , α2 , . . . , αn ] the n-th convergent to α. We observe that the distribution
pattern of the first p points of a van der Corput sequence generated from a
linear permutation with multiplier m is similar to the distribution pattern of
the first p points of a Kronecker or (kα) sequence, if m/p = [0; α1 , α2 , . . . , αn ]
is the n-th convergent of α for some n. Indeed, we can directly apply a theorem
of Niederreiter [12] on (kα) sequences, in which the irrational α has bounded
partial quotients, to obtain bounds for DN (Spσ ) with 1 ≤ N ≤ p; see also [15,
Corollary 3.5]. In a next step, we can apply the asymptotic method of Faure
[3,7], which uses the repetitive structure of van der Corput sequences, to obtain
Corollary 4.3 from the following (finite version of the) result of Niederreiter:
Theorem 4.2 For a prime base p and a multiplier 1 ≤ m ≤ p − 1, let σ(i) =
m · i (mod p). Set m/p = [a1 , . . . , an ] and let amax be such that ai ≤ amax for
all i. Then,
amax + 1
log(N + 1),
(4)
DN (Spσ ) ≤
log(amax + 1)
for all 1 ≤ N ≤ p.
Note that only the initial segment of a van der Corput sequence generated
from a linear permutation is similar to an (kα) sequence, hence we need to
invoke the results of Faure to obtain:
Corollary 4.3 With the same assumptions as in Theorem 4.2, we get
t(Spσ ) ≤ c6 ·
amax
,
log amax
f (Spσ ) ≤ c7 ·
a2max
.
log amax
(5)
This result agrees with the results of Faure and complements Theorem
4.1. If m = 1, then amax = p such that f (Spid ) < c7 · p2 / log p. Furthermore,
if 1 ≤ m ≤ (p − 1)/2 either divides p + 1 or p − 1, then we can set amax =
max{a, m}. It is important to note here that the distribution properties of
linear permutations generated from multipliers m1 and m2 are identical if
m1 + m2 = p.
Corollary 4.3 shows that an answer to our second question for the set of
linear permuations is intimately related to the question whether there exists
a multiplier m for every prime base p such that the partial quotients of m/p
are bounded by a constant independent of p. This question is known as the
Conjecture of Zaremba and is still open in its full generality. However, Bourgain and Kontorovich [2] recently announced major progress and confirmed
the conjecture for a subset of the integers of density 1.
It is interesting to note that the algorithm of Faure [8] to generate good
permutations in each integer base b heavily relies on a construction principle that is related to the original van der Corput sequence in base 2, while
Niederreiter proved [13] that Zaremba’s conjecture holds for all powers of 2
with amax = 3.
5
Conclusion
Computational results show that there always exist good multipliers for linear
permutations. Rather than picking multipliers randomly in applications, we
suggest to avoid multipliers that either divide p + 1 or p − 1. In fact, the best
multipliers are those with smallest bounded partial quotients.
Interestingly, our computational results also show that there always exist
permutations that behave considerably better than the best linear permutations in the same base [17]. This motivates future work on more general
classes of permutations that can be defined via permutation polynomials of
fixed Carlitz rank [1]; see also [19].
References
[1] Aksoy, E. , A. Cesmelioglu, W. Meidl, A. Topuzoglu, On the Carlitz rank of
permutation polynomials, Finite Fields Appl. 15 (2009), 428–440.
[2] Bourgain, J. and A. Kontorovich, On a conjecture of Zaremba, Manuscript
(2011), 96 pages, URL: http://arxiv.org/abs/1107.3776v1.
[3] Chaix, H. and H. Faure, Discrépance et diaphonie en dimension un, Acta Arith.
63 (1993), 103–141.
[4] Dick, J. and F. Pillichshammer, “Digital nets and sequences,” Cambridge
University Press, Cambridge, 2010.
[5] Erdös, P. and P. Turán, On a problem in the theory of uniform distribution. I,
Indag. Math. 10 (1948), 370–378.
[6] Erdös, P., Problems and results on diophantine approximations, Compos. Math.
16 (1964), 52–65.
[7] Faure, H., Discrépance de suites associées à un système de numération (en
dimension un), Bull. Soc. Math. France 109 (1981), 143–182.
[8] Faure, H., Good permutations for extreme discrepancy, J. Number Theory 42
(1992), 47–56.
[9] Faure, H., Discrepancy and diaphony of digital (0, 1)-sequences in prime base,
Acta Arith. 117 (2005), 125–148.
[10] Faure, H., Irregularities of distribution of digital (0, 1)-sequences in prime base,
Integers 5 (2005), no. 3.
[11] Koksma, J.F., Een algemeene stelling uit de theorie der gelijkmatige verdeeling
modulo 1, Mathematica B (Zutphen) 11 (1942/43), 7–11.
[12] Niederreiter, H., Applications of diophantine approximations to numerical
integration. In: Diophantine Approximation and Its Applications, C.F. Osgood
(Ed.), Academic Press, New York, 1973, 129-199.
[13] Niederreiter, H., Dyadic fractions with small partial quotients, Monatsh. Math.
101 (1986), 309–315.
[14] Niederreiter, H., Point sets and sequences with small discrepancy, Monatsh.
Math. 104 (1987), 273–337.
[15] Niederreiter, H., “Random Number Generation and Quasi-Monte Carlo
Methods,” SIAM, Philadelphia, 1992.
[16] Ostromoukhov, V., Recent progress in improvement of extreme discrepancy and
star discrepancy of one-dimensional Sequences, In: Monte Carlo and QuasiMonte Carlo Methods 2008, L’Ecuyer P. and A.B. Owen, Ed., Springer (2009),
561-572.
[17] Pausinger, F. and W. Ch. Schmid, A good permutation for one-dimensional
diaphony, Monte Carlo Methods Appl. 16 (2010), 307–322.
[18] Pausinger, F., Weak multipliers for generalized van der Corput sequences, J.
Théor. Nombres Bordeaux 24 (2012), no 3, 729–749.
[19] Pausinger, F. and A. Topuzoglu, Van der Corput sequences and permutation
polynomials, Preprint.
[20] Roth, K. F., On irregularities of distribution, Mathematika 1 (1954), 73–79.
[21] Schmidt, W.M., Irregularities of distribution. VII, Acta Arith. 21 (1972), 45–
50.
[22] Van der Corput, J.G., Verteilungsfunktionen. I, Proc. Kon. Ned. Akad. v.
Wetensch. 38 (1935), 813–821.
[23] Van der Corput, J.G., Verteilungsfunktionen. II, Proc. Kon. Ned. Akad. v.
Wetensch. 38 (1935), 1058–1066.
[24] Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77
(1916), 313–352.
[25] Zinterhof, P., Über einige Abschätzungen bei der Approximation von Funktionen
mit Gleichverteilungsmethoden, Österr. Akad. Wiss. SB II 185 (1976) 121–132.