THE DISCREPANCY OF GENERALIZED VAN-DER-CORPUT-HALTON SEQUENCES

THE DISCREPANCY OF GENERALIZED VAN-DER-CORPUT-HALTON SEQUENCES MICHAEL DRMOTA Abstract. The purpose of this paper is to provide upper bounds for the discrepancy of generalized Van-der-Corput-Halton sequences that are built from Halton sequences and the Zecendorf Van-der-Corput sequence.. Introduction Suppose that q 2 is a given integer and n = j 0 ε q,j (n)q j, ε q,j (n) {0,,..., q } denotes the q-ary representation of a non-negative integer n. Then the q-ary Vander-Corput (φ q (n)) n 0 sequence is defined by φ q (n) = j 0 ε q,j (n)q j. It is well nown that the Van-der-Corput sequence is a low discrepancy sequence, that is, the (star-) discrepancy D N (φ q(n)) satisfies DN(φ q (n)) log N N ; Similarly a d-dimensional Halton equences Φ q (n) that is defined by Φ q (n) = (φ q (n),..., φ qs (n)) satisfies D N(Φ q (n)) (log N)s N, if q = (q,..., q s ) consists of pairwise coprime integers q j 2. In particular all these sequences are uniformly distributed modulo (which just means that the discrepancy tends to 0 as N ). For more details on discrepancy theory and uniformly distributed sequences we refer to [2, 6]. We just note that the (star-) discrepancy DN (x n) of a s-dimensional real sequence (x n ) n 0 is defined by D N (x n ) = sup 0<α,...,α s N N [0,α) [0,α s)(x n mod ) α α s. Recently Hofer, Iaco, and Tichy [4] considered generalized Van-der-Corput and Halton sequences of the following form. Suppose that b and d 2 (or b 2 if d = ) are integers and G = (G n ) n 0 is an integer sequence defined by G 0 =, G = b(g + + G 0 ) + for < d and G = b(g + + G d ) for d. Then every non-negative integer n can be uniquely represented by (.) n = j 0 ε G,j (n) G j, Key words and phrases. Halton sequence, Zecendorf Van-der-Corput sequence, discrepancy.

2 MICHAEL DRMOTA where the digits ε {0,,..., G j+ /G j } are (uniquely) chosen by the greedy condition J ε G,j (n)g j < G J. j=0 We will call (.) the G-ary expansion of n. Of course, this generalizes q-ary expansions, where b = q, d =, and G = q. Furthermore let β > be the dominating root of the adjoint polynomial x d bx d b = 0. Then the (so-called) β-van-der-corput sequence (φ β (n)) n 0 is defined by φ β (n) = j 0 ε G,j (n)β j. As in the case of the usual Van-der-Corput sequence it is well nown that in this case the β-van-der-corput sequence is a low-discrepancy sequence, that is DN (φ β(n)) (log N)/N, see [, 8]. A very prominent example is the β-van-der-corput sequence corresponding to the golden mean ϕ = ( + 5)/2, where the base sequence G = (G n ) n 0 is given by the Fibonacci numbers G n = F n+2. Here the digital expansion (.) is called Zecendorf expansion. And the digits ε G,j (n) {0, } just have to satisfy ε G,j (n)ε G,j+ (n) = 0, that is, there are no consecutive s. For convenience we will denote this sequence the Zecendorf Van-der-Corput sequence (φ Z (n)) n 0. Similarly to the usual Halton sequence the β-halton sequence is defined by Φ β (n) = (φ β (n),..., φ βs (n)), where the entries of β = (β,..., β s ) correspond to G i -adic expansions of the above ind, i s. By the use of ergodic properties it was shown in [4] that Φ β (n) is uniformly distributed modulo provided that the integers b i are pairwise coprime and the dominant roots β i have the property that βi /βl j Q for all integers, l and i j. However, the discrepancy was not considered at all. The purpose of the present paper is to provide a first quantitative discrepancy analysis of β-halton sequences. Theorem. Suppose that q,..., q s 2 are pairwise coprime integers. Then the discrepancy of the (s + )-dimensional sequence satisfies for every ε > 0. Φ(n) = (φ q (n),..., φ qs (x), φ Z (n)) D N (Φ(n)) N ε It remains an open problem whether this ind of generalized Halton sequences are low discrepancy sequences. Nevertheless the upper bound given in Theorem is close to optimality. We leave this as an open problem. Problem. Suppose that Φ β (n) = (φ β (n),..., φ βs (n)) is a s-dimensional β-halton sequence that is uniformly distributed modulo. Is Φ β (n) also a low-discrepancy sequence, too, that is, DN (log N)s (Φ β (n))? N Actually it is not clear how far Theorem can be generalized. It would be desirable to cover (at least) the ind of sequences that are discussed in [4]. However, it seems that the methods that are applied in the present paper are not sufficient to handle these cases. In the case of the Zecendorf Van-der-Corput sequence the distribution can be reduced to distribution properties of the Weyl sequence nϕ mod (see Lemma 5). This is due to the fact that the Zecendorf expansion agrees with the Ostrowsi expansion related to the golden mean ϕ. In the more general

THE DISCREPANCY OF GENERALIZED VAN-DER-CORPUT-HALTON SEQUENCES 3 case it might be possible to replace this approach by a distribution analysis of a more-dimensional linear sequence (modulo ) in Rauzy fractal type sets (see [7, ]). 2. Discrepancy Bounds for Halton Sequences The purpose of this section is to present a very basic approach to the discrepancy of Halton sequences Φ q (n). These results are by no means new but are helpful to prepare the proof of Theorem. Actually all subsequent properties follow from the following observation (that is immediate from the definition). Lemma 2. Suppose that q 2, 0, and 0 m < q are integers. Then we have [ m φ q (n) q, m + ) q if and only if ε q, l (n)q l = m, l=0 that is, the digits ε q,0 (n),..., ε q, (n) are fixed or, equivalently, n is contained in a fixed residue class mod q. Lemma 2 implies a discrepancy bound (2.) for the Van-der-Corput sequence. Note that in the present case we trivially have δ so that the upper bound q L + L/N (log N)/N follows immediately by choosing L = log q N. The reason for using the formulation (2.) with explicit δ is that this ind of formula naturally generalizes to (generalized) Halton sequences. Lemma 3. Suppose that q 2 is an integer. Then we have for every integer L (2.) D N(φ q (n)) q L + N L δ, where δ = max 0 u<q #{n < N : n u mod q } N q Proof. Suppose that the interval [0, α) = I I 2 I R is partitioned into R disjoint intervals I r of lengths l r, r R. Then by the triangle inequality we have N R [0,α) (x n ) Nα N Ir (x n ) Nl r. Now if α [vq L, (v + )q L ) and v has the digital expansion v = v 0 + v q + + v L q L then we can partition the interval [0, α) into v 0 + v + + v L + intervals: v 0 intervals of the form [m 0 q, (m 0 + )q ), 0 m 0 < v ; v intervals of the form [v 0 q + m q 2, v 0 q + (m + )q 2 ), 0 m < v etc., and finally the interval [vq L, α). By Lemma 2 it follows that for every interval of the form I = [mq, (m+)q ) we have N I (φ q (n)) Nq δ. r=

4 MICHAEL DRMOTA Finally for the interval [vq L, α) we set J = [vq L, (v + )q L ) and obtain N [vq L,α)(φ q (n)) N J (φ q (n)) N J (φ q (n)) Nq L + Nq L δ L + Nq L. and consequently N [vq L,α)(φ q (n)) (α vq L )N δ L + 2Nq L. Of course this proves Lemma 3. By using precisely the same proof method as in the proof of Lemma 3 we obtain a direct generalization for Halton sequences. Note that the Lemma 2 together with the Chinese remainder theorem has to be used to obtain (2.2). Note again that δ,..., s so that we derive from (2.2) the upper bound D N (Φ q(n)) (log N) s /N by choosing L j = log qj N. Lemma 4. Suppose that q, q 2,..., q s 2 are pairwise coprime integers and q = (q,..., q s ). Then we have for all integers L,..., L s s (2.2) DN (Φ q (n)) + δ,... j= q Lj N s j L s L s with δ,..., s = max 0 u<q qs s #{n < N : n u mod q qs s } N qs s Finally we mention that it is easy to we extend Lemma 4 to subsequences of Halton sequences. Suppose (again) that q, q 2,..., q s 2 are pairwise coprime integers. If c(n) a sequence of non-negative integers then we obtain s with D N (Φ q (c(n))) δ,..., s = max 0 u<q qs s j= q Lj j + N L q s L s δ,... s #{n < N : c(n) u mod q qs s } N qs s Similar observations have been already made in [5]. And actually we can reprove them with the above estimates. We just recall the obervation from [5] that (Φ q (c(n))) n 0 is uniformly distributed modulo if and only if c(n) is uniformly distributed in the residue classes modulo (q q s ) for all 0. (Of course if c(n) is uniformly distributed in the residue classes modulo (q q s ) then it is also uniformly distributed in the residue classes modulo q qs s for all j.) It is also of interest to start with a sequence r(n) of non-negative real numbers and to consider the sequence (Φ q ( r(n) ) n 0. Here it follows that if the sequence r(n)/(q qs s ) is uniformly distributed modulo for all integers,... s 0 then (Φ q ( r(n) ) n 0 is uniformly distributed modulo, too. Furthermore D N(Φ q ( r(n) ) s j= q Lj j q + D N (r(n)/(q qs s )). L s L s

THE DISCREPANCY OF GENERALIZED VAN-DER-CORPUT-HALTON SEQUENCES 5 3. Discrepancy Bounds for the Zecendorf Van-der-Corput Sequence Next we provide a quantitive appraoch to the Zecendorf Van-der-Corput Sequence (φ Z (n)) n 0 that has strong similarities to the q-ary case from the previous section. We recall that the Fibonacci number (F n ) n 0 are given by F 0 = 0, F =, and F = F + F 2 (for 2) and that every non-negative integer n 0 has a unique representation n = j 2 ε Z,j (n)f j, where ε Z,j (n) {0, } and ε Z,j (n)ε Z,j+ (n) = 0. The Zecendorf Van-der-Corput Sequence φ Z (n) is then given by φ Z (n) = j 2 ε Z,j (n)ϕ j+, where ϕ = ( + 5)/2 is the golden mean. There is an interesting analogue to Lemma 2. information than in the q-adic case. Actually we get slightly more Lemma 5. Suppose that 3 and that the first digits ε Z,2 (n),..., ε Z, (n) are fixed. Then we equivalently have [ ) µ, µ+ if ε ϕ (3.) φ Z (n) 2 ϕ 2 Z, (n) = 0, [ ) µ, µ+ϕ if ε ϕ 2 ϕ 2 Z, (n) =, where Furthermore we have 3 µ = ε Z, l (n)ϕ l. l=0 (3.2) ( ) nϕ ( ) uϕ + where and [ A (0) = ) ϕ, ϕ { A (0) + Z if ε Z, (n) = 0, A () + Z if ε Z, (n) =, u = ε Z,j (n)f l j=2 [ and A () = ) ϕ +, ϕ. Proof. Both properties, (3.) and (3.2), are completely elementary. We just mention that (3.2) is a general property related to the Ostrowsi expansion, compare with [3, Section 3.2]. (For a proof in the present case we refer to [0].) [ ) [ We first note that the intervals µ, µ+ (or µ, ), µ+ϕ respectively) partition the unit interval [0, ) if the digits ε Z,2 (n),..., ε Z, (n) vary over all valid ϕ 2 ϕ 2 ϕ 2 ϕ 2 0--sequences. Similarly the sets ( )uϕ + A (0) mod (or ( )uϕ + A () mod, respectively) partition the unit interval. This implies that the distribution of the Zecendorf Van-der-Corput sequence φ Z (n) can be directly related to the distibution of the sequence nϕ mod. This leads us to a corresponding variant of Lemma 3. Lemma 6. For every integer L we have D N (φ Z (n)) ϕ L + N L δ,

6 MICHAEL DRMOTA where δ = sup 0 β< N [β,β+ϕ )(ϕn mod ) N ϕ. Proof. The proof is very close to the proof of Lemma 3. First, for every α (0, ] there exists µ of the form L 3 µ = ε L l ϕ l. l=0 with ε 2,..., ε L {0, } and ε j ε j+ = 0 (that is, there are no consecutive s) such that µϕ L+2 α < (µ+)ϕ L+2 if ε L = 0 or µϕ L+2 α < (µ+ϕ)ϕ L+2 if ε L =. For notational convenience we only tae into account non-zero digits and write L µ = ϕ L 2 lj, j= where L L/2 and l < l 2 < < l L L 2. Now we partition the interval [0, α) into L + intervals of the form [ ) [ 0, ϕ l ϕ, l ϕ + ) L L l ϕ l2 ϕ, L lj ϕ lj ϕ, α lj The first L intervals can be seen as intervals of the form given in (3.). Let I denote one of these intervals. We now apply Lemma 5 and by (3.2) there is another interval J (mod ) of the same length ϕ lj such that N I (φ Z (n)) = N j= j= J (ϕn mod ). Thus the local discrepancy of the sequence φ Z (n) with respect to the interval I can be replaced by the local discrepancy of the sequence ϕn mod with respect to the interval J. And the second one can be estimated by δ lj. Finally the remaining interval can be handled in the same was in the proof of Lemma 3. Summing up this leads to proposed discrepancy bound. We should add that the sets [β, β + ϕ ) are bounded remainder sets for the sequence ϕn mod since ϕ Z + ϕz and we also have δ = O(). This leads to another proof of the upper bound DN(φ Z (n)) log N N. We also want to mention that Lemma 5 extends to subsequences. Suppose that c(n) is a sequence of non-negative integers. Then we have for every integer L where δ = In particular we get D N(φ Z (c(n))) ϕ L + N sup 0 β< L δ, N [β,β+ϕ )(ϕc(n) mod ) N ϕ. D N(φ Z (c(n))) log N D N (ϕc(n)). For example it follows from Lemma 6 that φ Z (p(n)) or φ Z (p n ) is uniformly distributed mod, where p(n) is a non-negative integer valued polynomial and p n is the sequence of primes. j=

THE DISCREPANCY OF GENERALIZED VAN-DER-CORPUT-HALTON SEQUENCES 7 Finally the method of Lemma 5 can be used to describe the joint distribution of a Halton sequence and the Zecendorf Van-der-Corput sequence. Lemma 7. Suppose that q, q 2,..., q s 2 are pairwise coprime integers and q = (q,..., q s ). Then we have for all integers L, L,..., L s DN(Φ q (n), φ Z (n)) s ϕ L + + δ,,... j= q Lj N s j L L s L s with δ,,..., s = sup 0 β< N/(q qs s ) Proof. Suppose that ε Z, = 0. Then [ ) m (Φ q (n), φ Z (n)), m + q q [β,β+ϕ )(ϕq qs s n mod ) [ m q N ϕ q qs s. ), m [ + µ q ϕ 2, µ + ) ϕ 2 if any only if n contained in a residue class modulo q qs s and ϕn mod is contained in an interval of length ϕ +2. Thus, the number of n < N with this property minus the expected number N/(ϕ q qs s ) is bounded by δ 2,,..., s. (Similarly if ε Z, = then the upper bound is δ,,..., s.) Hence, by using a decomposition of a (d + )-dimensional interval [0, α ) [0, α s ) [0, α) is in the proofs of Lemma 3 and Lemma 6 we immediately obtain the result. 4. Proof of Theorem The proof of Theorem is based on Lemma 7. So we have to estimate δ,,..., s. Clearly we have 2 (N/q) DN/q (ϕqn), δ,,..., s where q = q qs s. It is well nown that the discrepancy of a sequence x n can be estimated by the inequality of Erdős-Turán (see [2, 6]) saying that for every integer H D M (x n ) H + H h= h M M If x n = αn for some irrational number α we have M e 2πihαn hα, e 2πihxn. where x = min Z x denotes the distance to the integers. Consequently we get (for every integer H ) (4.) δ,,..., s N qh + H h= h hqϕ. In order to handle this ind of sums we mae use of Ridout s p-adic version of the Thue-Siegel-Roth theorem [9] which implies the following lemma. Lemma 8. Suppose that q, q 2,..., q s 2 are pairwise coprime integers. Then for every ε > 0 there exists a constant C > 0 such that for all integers,..., s 0 and h q qs s hϕ C h +ε (q qs s ) ε

8 MICHAEL DRMOTA Proof. Set q = q qs s and let D denote the set of primes that appear in the prime decomposition of q,..., q s. By Ridoux s theorem for every ε > 0 there exist a constant C > 0 with qhϕ l D for all integers q, h. Since we obtain as proposed. l D qh l qh l l D qhϕ C (qh) +ε q l = q C h +ε q ε Lemma 9. Suppose that q, q 2,..., q s 2 are pairwise coprime integers. Then for every ε > 0 there exists a constant C > 0 such that for all integers,..., s 0 and H H h= h q qs s hϕ C(q qs s H) ε. Proof. Let Q = Q (α) denote the denominators of the convergents P /Q of an irrational number α and suppose that Q < H Q. Then it follows from the approximation that α = P Q + θ Q K Q K+, θ, hα hp <, h Q. Q Q Since P and Q are coprime the numbers hp, h Q, run through all residue classes modulo Q. Thus, the numbers hα, h Q, can be well approximated by l/q with l {0,,..., Q } but with at most three exceptions that are related to l {0,, Q }. αh For these exceptional values we can just say that min hα whereas for the other values we have αh l/q /Q. h H Consequently we obtain H h= hα 3 min hα + h H Q 2 l=2 min h H hα + Q log Q l/q /Q If we apply this procedure for α = qϕ (with q = q qs s ) then we can use the estimate from Lemma 8 to obtain min h H hqϕ H+ε q ε. Furthermore, since Q α /Q we obtain (again from Lemma 8) C Q qϕ Q Q +ε qε H +ε q ε and consequently H hqϕ H+ε q ε + H +ε q ε log(h +ε q ε ) H +2ε q 2ε. h=

THE DISCREPANCY OF GENERALIZED VAN-DER-CORPUT-HALTON SEQUENCES 9 Finally by partial summation we obtain H h hqϕ H h= H h= (qh) 2ε. H hqϕ + h 2 h= h l= lqϕ Since ε > 0 is arbitrary we can replace 2ε by ε and we are done. Now it is easy to complete the proof of Theorem. Proof. We use Lemma 7, where we choose L = log ϕ N and L j = log qj N. As above we abbreviate q qs s by q. We distinguish two cases. First suppose that q > N. Then we trivially have δ,,..., s 2. In the other case we set H = N/q and apply (4.) to obtain Summing up this implies δ,,..., s N qh + H + N ε N ε. h= h hqϕ DN(Φ(n)) N + N N ε (log N) s+ N 2ε. Again, since ε > 0 is arbitrary we can replace 2ε by ε which completes the proof of Theorem. Acnowledgements. The author thans Luas Spiegelhofer and Wolfgang Steiner for several intersting discussions on the topic of the paper and Yann Bugeaud for suggesting the use of Ridout s theorem. This research was supported by the Austrian Science Foundation FWF, grant S 5502 that is part of the SFB Quasi-Monte Carlo Methods: Theory and Applications. References [] G. Barat and P. Grabner, Distribution properties of g-additive functions, Journal of Number Theory 60 (996), 03 23. [2] M. Drmota and R. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Mathematics 65, Springer, 997. [3] A. Haynes, Equivalence classes of codimension one cut-and-project sets, Ergodic Theory Dyn. Syst., to appear. arxiv:3.7277 [4] Marus Hofer, Maria Rita Iaco and Robert Tichy, Ergodic properties of β-adic Halton sequences, Ergodic theory and dynamical systems 35 (3), (205), 895 909. [5] R. Hofer, P. Kritzer, G. Larcher, F. Pillichshammer Distribution properties of generalized van der Corput-Halton sequences and their subsequences. Int. J. Number Theory 4 (5), (2009), 79 746. [6] Kuipers and H. Niedereiter, Uniform Distribution of Sequences, Wiley, New Yor, 974. [7] M. Minervino and W. Steiner, Tilings for Pisot beta numeration, Indagationes Mathematicae 25 (4) (204), 745 773. [8] S. Ninomiya, Constructing a new class of low-discrepancy sequences by using the β-adic transformation, Math. Comput. Simulation 47(2), 403 48. [9] D. Ridout, Rational approximations to algebraic numbers, Mathematia 4 (957), 25 3. [0] L. Spiegelhofer, Correlations for Numeration Systems, PhD thesis, TU Wien, 204. [] W. Steiner, Parry expansions of polynomial sequences, INTEGERS: The Electronic Journal of Combinatorial Number Theory 2 (2002), A4, 28 pages.

0 MICHAEL DRMOTA E-mail address: michael.drmota@tuwien.ac.at Institut für Disrete Mathemati und Geometrie TU Wien, Wiedner Hauptstr. 8 0, 040 Wien, Austria