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Monatshefte für Mathematik

, Volume 188, Issue 1, pp 1–29 | Cite as

The generalized and modified Halton sequences in Cantor bases

  • Dušan Bednařík
  • Poj LertchoosakulEmail author
  • Diego Marques
  • Pavel Trojovský
Open Access
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Abstract

This paper aims to generalize results that have appeared in Atanassov (Math Balk New Ser 18(1–2):15–32, 2004). We consider here variants of the Halton sequences in a generalized numeration system, called the Cantor expansion, with respect to arbitrary sequences of permutations of the Cantor base. We first show that they provide a wealth of low-discrepancy sequences by giving an estimate of (star) discrepancy bound of the generalized Halton sequence in bounded Cantor bases. Then we impose certain conditions on the sequences of permutations of the Cantor base which are analogous, but not straightforward, to the modified Halton sequence introduced by E.I. Atanassov. We show that this modified Halton sequence in Cantor bases attains a better estimate of the (star) discrepancy bound than the generalized Halton sequence in Cantor bases.

Keywords

Halton sequence van der Corput sequence Hammersley point set Low-discrepancy sequence Pseudorandom number Cantor expansion 

Mathematics Subject Classification

Primary 11J71 11K38 11K45 Secondary 65C10 

1 Introduction

Let \(\omega = (x_n)_{n=1}^\infty \) be a sequence in \([0,1)^s\). A standard problem in numerical analysis is estimating the integral of a function, through a knowledge of its value at a finite number of points of the sequence. This is known as the Monte Carlo method in the case of stochastic sequences \((x_n)_{n=1}^N\) or the quasi-Monte Carlo method in the case of deterministic \((x_n)_{n=1}^N.\) This is encapsulated in the famous Koksma–Hlawka inequality
$$\begin{aligned} \bigg | \int _{[0,1]^s} f(x)\, dx - \frac{1}{N}\sum _{n=1}^N f(x_n)\bigg | \le V(f) D^*_N(\omega ) \end{aligned}$$
for any function f on \([0,1]^s\) with bounded variation V(f) in the sense of Hardy and Krause, see [19], and for any finite set of points \((x_n)_{n=1}^N\) with discrepancy
$$\begin{aligned} D^*_N(\omega ) = \sup _{J = \prod _{i=1}^s [0,z_i) \subseteq [0,1)^s}\bigg | \frac{A(J;N;\omega )}{N}- \lambda _s(J)\bigg |. \end{aligned}$$
Here \(A(J;N;\omega ) = \#\{ 1\le n\le N:x_n\in J\}\) is the counting function, \(\lambda _s(J)\) denotes the s-dimensional Lebesgue measure of J,  and the above supremum is taken over all rectangular solids \(J=\prod _{i=1}^s [0,z_i)\) with \(0< z_i \le 1\) \((1\le i\le s)\). Note that \(\lambda _s(J) = \prod _{i=1}^s z_i.\) For more details on numerical integration, the reader can consult [5, 15] or [16]. Evidently, to estimate \(\int _{[0,1]^s} f(x)\,dx\) sufficiently precisely, what is needed is a good bound for \(D^*_N(\omega ).\) The discrepancy is nothing other than a quantitative measure of uniformity of distribution. In particular, the sequence \(\omega \) is uniformly distributed on \([0,1)^s\), if and only if \(D^*_N(\omega )\rightarrow 0\) as \(N\rightarrow \infty .\) In a sense, the faster \(D^*_N(\omega )\) decays as a function of N,  the better uniformly distributed the sequence \(\omega \) is. One of the fundamental obstructions in nature in this subject is that there is a limit to how well distributed any sequence can be. This is encapsulated in the elementary inequality \(D^*_N(\omega ) \ge 1/2^sN\) \((N\in {\mathbb {N}})\) whose proof makes an entertaining exercise. This opens the door to the deep subject of irregularities of distribution which addresses just what limitations there are to the uniformity of distribution of an arbitrary sequence, and the complementary problem of constructing sequences with discrepancy as small as possible. This latter issue is clearly central to the initial issue mentioned in this paper.
Perhaps the most famous example of a low-discrepancy sequence is the van der Corput sequence. In 1935, van der Corput [4] introduced a procedure to generate low-discrepancy sequences on [0, 1). These sequences are considered to be among the best distributed over [0, 1),  and no other infinitely generated sequences can have discrepancy of smaller order of magnitude than van der Corput sequences. The technique of van der Corput is based on a very simple idea. Let \(b>1\) be a natural number. Then every nonnegative integer n has a unique b-adic representation of the form
$$\begin{aligned} n = \sum _{j=1}^\infty n_jb^{j-1} = n_1+n_2b+n_3b^2+n_4b^3+\cdots , \end{aligned}$$
where \(n_j\in \{0,1,\cdots , b-1\}\) \((j\in {\mathbb {N}})\) and only finitely many of the \(n_j\)’s are nonzero. The van der Corput sequence \((\phi _b(n))_{n=0}^\infty \) in base b is constructed by reversing the base b representation of the sequence of nonnegative integers, where the radical-inverse function \(\phi _b:{\mathbb {N}}_0\rightarrow [0,1)\) is defined by
$$\begin{aligned} \phi _b\left( \sum _{j=1}^\infty n_jb^{j-1}\right) = \sum _{j=1}^\infty \frac{n_j}{b^j} = \frac{n_1}{b}+\frac{n_2}{b^2} +\frac{n_3}{b^3}+\cdots . \end{aligned}$$
In applications, a generalization of the van der Corput sequence to higher dimensions is more likely to be of practical use. In 1960, this was proposed by J.H. Halton [11]. Given pairwise coprime integers \(b_1,\cdots , b_s\) all greater than 1,  the sequence \((\phi _{b_1}(n),\cdots ,\phi _{b_s}(n))_{n=0}^\infty \) is called the Halton sequence in bases \(b_1,\cdots , b_s.\)
It was known for a long time that the discrepancy of the first N elements of the Halton sequence in bases \(b_1,\cdots , b_s\) can be bounded by
$$\begin{aligned} c\frac{(\log N)^s}{N} + O\bigg (\frac{(\log N)^{s-1}}{N}\bigg ), \end{aligned}$$
(1)
for some constant \(c=c(b_1,\cdots , b_s)>0\). For example, this was shown in [9, 11, 18, 19]. It is believed that the order \((\log N)^s/N\) is the best possible for an arbitrary infinite sequence. That this is the case when \(s=1\) was proved by Schmidt [21]. For \(s>1,\) the question remains open. We shall call an infinite sequence \(\omega \) in \([0,1)^s\) a low-discrepancy sequence if \(D^*_N(\omega ) = O((\log N)^s/N).\)
The question of how small the constant c in (1) can be is interesting from both a theoretical and a practical viewpoint. The articles referred to above show that this constant depends very strongly on the dimension s. The minimal value for this quantity can be obtained if we choose \(b_1,\cdots , b_s\) to be the first s prime numbers. But even in this case, c grows very fast to infinity if s increases. This deficiency was overcome by Atanassov [1] who could improve the constant so that
$$\begin{aligned} c = c(b_1,\cdots , b_s) = \frac{1}{s!} \prod _{i=1}^s \frac{b_i -1}{\log b_i}. \end{aligned}$$
(2)
This estimate is so impressive that, when \(b_1,\cdots , b_s\) are the first s prime numbers, \(c(b_1,\cdots , b_s)\rightarrow 0\) as \(s\rightarrow \infty .\)
In another direction of effort to improve the behavior of Halton sequences, several researchers have studied various ways of generalizing their definition by including permutations, chosen either deterministically or randomly, in the radical-inverse function. This idea goes back to [2, 6]. Let \(\varSigma = (\sigma _j)_{j=1}^\infty \) be an arbitrary sequence of permutations of \(\{0,1,\cdots , b-1\}\) which fix 0. The generalized radical-inverse function \(\phi _b^\varSigma :{\mathbb {N}}_0\rightarrow [0,1)\) with respect to \(\varSigma \) is defined by
$$\begin{aligned} \phi _b^\varSigma \left( \sum _{j=1}^\infty n_j b^{j-1}\right) = \sum _{j=1}^\infty \frac{\sigma _j(n_j)}{b^j} = \frac{\sigma _1(n_1)}{b}+\frac{\sigma _2(n_2)}{b^2} + \frac{\sigma _3(n_3)}{b^3} +\cdots . \end{aligned}$$
The sequence \((\phi _b^\varSigma (n))_{n=0}^\infty \) is a low-discrepancy sequence, and it is called the generalized van der Corput sequence in base b with respect to \(\varSigma .\) The generalized Halton sequences can be introduced in a similar way. In parallel to these efforts, Atanassov also showed in [1] that any generalized Halton sequence attains the same constant as in (2); furthermore, he could produce certain generalized Halton sequences, by means of the so-called “admissible integers,” for which the constants \(c=c(b_1,\cdots , b_s,\varSigma _1,\cdots , \varSigma _s)\) of the discrepancy bounds have an even better asymptotic behavior than (2).

In this paper, we introduce the generalized Halton sequence in Cantor bases, which is induced by the a-adic integers and which is called the Cantor expansion, and give an estimate of its discrepancy by adapting the techniques developed by Atanassov. Also, we extend the notion of admissible integers so that we can derive a special type of generalized Halton sequences in Cantor bases with a better estimate of discrepancy bounds. Our work is an extension of [10] and can be viewed as a generalization of Atanassov’s results. Note that the van der Corput sequence and some other one-dimensional low-discrepancy sequences with respect to the Cantor expansion were studied in [3, 8]. In addition, Halton sequences defined in a more generalized numeration system than the Cantor expansion, called the G-expansion, were mentioned in [13]; however, the paper aimed to study the Halton sequence in some fixed non-integer bases. Furthermore, it is worth noting that several uniformly distributed sequences, which can be constructed through the notions of Cantor-base-additive function and strongly Cantor-base-additive function, were studied in [14]. This paper also included our generalized Halton sequence in Cantor bases as an example; nevertheless, it aimed to provide criteria for uniform distribution and it did not study the discrepancy of those sequences obtained by Cantor-base-additive functions.

We now summarize the contents of this paper. In Sect. 2, we introduce the concept of a generalized numeration system, called the Cantor expansion. Then we define the generalized Halton sequence induced by this generalized system and state our first main result on the estimate of discrepancy of the sequence. In Sect. 3, we impose certain conditions on the sequences of permutations of the Cantor base to produce an extension of the concept of admissible integers. Then we define the modified Halton sequence in the Cantor expansion and state our second main result regarding the discrepancy bound of this special type of sequence. In Sects. 4 and 5, we prove the first and the second main results, respectively. Finally, we introduce in Sect. 6 the generalized Hammersley point set in Cantor bases and show that it provides a wealth of low-discrepancy point sets by giving an estimate of its discrepancy.

We list here the notation which will be used repeatedly throughout the paper. For each natural number \(b>1,\) we write \({\mathbb {Z}}_b = \{0,1,\cdots , b-1\}\) and \({\mathbb {Z}}_b^* = \{1,2,\cdots , b-1\}.\) It is also important to note that every permutation in this paper fixes 0.

2 The generalized Halton sequence in Cantor bases

Let \(b = (b_j)_{j=1}^\infty \) be a sequence of natural numbers greater than 1. Then it is clear that every nonnegative integer n has a unique b-adic representation of the form
$$\begin{aligned} n = \sum _{j=1}^\infty n_jb_1\cdots b_{j-1} = n_1 + n_2b_1 + n_3b_1b_2 + n_4b_1b_2b_3 + \cdots , \end{aligned}$$
where \(n_j\in {\mathbb {Z}}_{b_j}\) \((j\in {\mathbb {N}})\) and all but finitely many \(n_j\)’s are zero. This b-adic representation is also called the Cantor expansion of n with respect to the Cantor base b. Moreover, every real number \(x\in [0,1)\) has a b-adic expansion of the form
$$\begin{aligned} x = \sum _{j=1}^\infty \frac{x_j}{b_1\cdots b_j} = \frac{x_1}{b_1} +\frac{x_2}{b_1b_2} + \frac{x_3}{b_1b_2b_3} + \cdots , \end{aligned}$$
where \(x_j\in {\mathbb {Z}}_{b_j}\) \((j\in {\mathbb {N}}).\) The \(x_j\) can be calculated by the greedy algorithm
$$\begin{aligned} x_1 = [xb_1] \quad \text{ and }\quad x_j = [\{xb_1\cdots b_{j-1}\}b_j], \end{aligned}$$
where \([\alpha ]\) and \(\{\alpha \}\) denote the integer part and the fraction part of \(\alpha ,\) respectively. The idea of this generalized numeration system stems from the a-adic integers, which is a class of locally compact topological groups and possesses a symbolic dynamical structure. For more details on the a-adic integers, see [12, pp. 106–117].
Suppose that \(\varSigma = (\sigma _j)_{j=1}^\infty \) is a sequence of permutations of \({\mathbb {Z}}_{b_1}, {\mathbb {Z}}_{b_2},{\mathbb {Z}}_{b_3},\cdots ,\) where the permutations all fix 0. We define the generalized radical-inverse function \(\phi _{b}^\varSigma :{\mathbb {N}}_0\rightarrow [0,1)\) by
$$\begin{aligned} \phi _{b}^{\varSigma } \left( \sum _{j=1}^\infty n_jb_1\cdots b_{j-1}\right) = \sum _{j=1}^\infty \frac{\sigma _j(n_j)}{b_1\cdots b_j} = \frac{\sigma _1(n_1)}{b_1} + \frac{\sigma _2(n_2)}{b_1b_2} + \frac{\sigma _3(n_3)}{b_1b_2b_3} + \cdots . \end{aligned}$$
The generalized van der Corput sequence in base b with respect to \(\varSigma \) is defined as \((\phi _{b}^{\varSigma }(n))_{n=0}^\infty .\) This sequence was studied in [3, 8], where it was proved to be a low-discrepancy sequence with some restriction on the Cantor base b. Furthermore, the sequence where all the permutations are identity was shown, without any restriction on the Cantor base, to be uniformly distributed mod 1 in [17] and to be a low-discrepancy sequence in [10].

Let \(b_1 = (b_{1,j})_{j=1}^\infty ,\cdots , b_s = (b_{s,j})_{j=1}^\infty \) be s sequences of natural numbers greater than 1 such that, for all \(1\le i_1<i_2\le s\) and all \(j_1,j_2\in {\mathbb {N}},\) \(b_{i_1,j_1}\) and \(b_{i_2,j_2}\) are coprime. For each \(1\le i\le s,\) let \(\varSigma _i = (\sigma _{i,j})_{j=1}^\infty \) be a sequence of permutations of \({\mathbb {Z}}_{b_{i,1}},{\mathbb {Z}}_{b_{i,2}},{\mathbb {Z}}_{b_{i,3}},\cdots .\) The generalized Halton sequence in Cantor bases \(b_1,\cdots , b_s\) with respect to \(\varSigma _1,\cdots , \varSigma _s\) is defined to be \((\phi _{b_1}^{\varSigma _1}(n),\cdots , \phi _{b_s}^{\varSigma _s}(n))_{n=0}^\infty .\)

The following theorem is our first main result which gives an estimate of discrepancy of the generalized Halton sequence in bounded Cantor bases.

Theorem 1

Let \(b_1 = (b_{1,j})_{j=1}^\infty , \cdots , b_s = (b_{s,j})_{j=1}^\infty \) be s bounded sequences of natural numbers greater than 1 such that, for all \(1\le i_1 < i_2 \le s\) and all \(j_1,j_2\in {\mathbb {N}},\) \(b_{i_1,j_1}\) and \(b_{i_2,j_2}\) are coprime. For each \(1\le i\le s\) and \(j\in {\mathbb {N}},\) let \(\sigma _{i,j}\) be a permutation of \({\mathbb {Z}}_{b_{i,j}}.\) For each \(1\le i\le s,\) denote \(\varSigma _i = (\sigma _{i,j})_{j=1}^\infty .\) Suppose that \(\omega \) is the generalized Halton sequence in Cantor bases \(b_1,\cdots ,b_s\) with respect to \(\varSigma _1,\cdots ,\varSigma _s.\) Then, for any \(N\in {\mathbb {N}},\) we have
$$\begin{aligned} N D^*_N(\omega ) \le \sum _{l =0}^{s} \frac{M_{l+1}}{l!} \prod _{i=1}^l\left( \frac{\lfloor M_i/2\rfloor \log N}{\log m_i} + l\right) , \end{aligned}$$
where \(M_i = \max (b_{i,j})_{j=1}^\infty \) and \(m_i = \min (b_{i,j})_{j=1}^\infty \) \((1\le i\le s),\) and where \(M_{s+1} = 1.\) In particular, for any \(N\in {\mathbb {N}},\) we obtain
$$\begin{aligned} D^*_N(\omega ) \le c \frac{(\log N)^s}{N} + O\left( \frac{(\log N)^{s-1}}{N}\right) \end{aligned}$$
with
$$\begin{aligned} c=c(b_1,\cdots , b_s) = \frac{1}{s!}\prod _{i=1}^s \frac{\lfloor M_i/2\rfloor }{\log m_i}. \end{aligned}$$

This theorem says that the generalized Halton sequence in bounded Cantor bases is a low-discrepancy sequence. In particular, it generalizes the main result in [10, Main Theorem 2.1 and Corollary  2.2], where all the permutations are fixed to be identity. Also, the constant \(c=c(b_1,\cdots , b_s)\) in the bound here is essentially as good as that established in [10].

When the sequences \(b_1,\cdots , b_s\) are of period one, that is, \(M_i=m_i\) for all \(1\le i\le s,\) the estimated bound \(c=c(b_1,\cdots ,b_s)\) of Theorem 1 is exactly the same as that given in [1] for the Halton sequences based on coprime bases. Though the generalized Halton sequence in Cantor bases do not attain a lower estimate of discrepancy bound than the classical Halton sequence, it provides more variety of sequences with similar estimated bound, especially when \(M_i\) is large and when the difference \(M_i-m_i\) is small compared with \(M_i\) for each \(1\le i <s.\) In fact, let \(c=c(b_1,\cdots ,b_s)\) and \(c'=c'(b_1',\cdots , b_s')\) denote the constants of the estimated bound appeared in Theorem 1 for the generalized Halton sequence in Cantor bases \(b_1,\cdots , b_s\) with respect to \(\varSigma _1,\cdots ,\varSigma _s\) and for the classical Halton sequence in bases \(b_1',\cdots , b_s',\) respectively, such that \(M_i=\max (b_{i,j})_{j=1}^\infty = b_i'\) for each \(1\le i\le s.\) Suppose that, for each \(1\le i\le s,\) \(k_i\in {\mathbb {N}}_0\) is a fixed integer and that \(M_i = m_i+k_i,\) where \(m_i = \min (b_{i,j})_{j=1}^\infty .\) Then we have
$$\begin{aligned} \frac{c(b_1,\cdots ,b_s)}{c'(b_1',\cdots , b_s')} = \frac{\frac{1}{s!}\prod _{i=1}^s \frac{\lfloor M_i/2\rfloor }{\log m_i}}{\frac{1}{s!}\prod _{i=1}^s \frac{\lfloor M_i/2\rfloor }{\log M_i}} = \prod _{i=1}^s \frac{\log M_i}{\log m_i} = \prod _{i=1}^s \frac{\log M_i}{\log (M_i-k_i)}. \end{aligned}$$
It is not hard to see that \(c(b_1,\cdots ,b_s)/c'(b_1',\cdots , b_s')\) tends to 1 exponentially fast as \(M_i\)’s go to infinity.

3 The modified Halton sequence in Cantor bases

In this section, we introduce a special class of generalized Halton sequences in Cantor bases that involves some deep periodicity properties. It can be considered as a generalization of Atanassov’s modified Halton sequences. We shall show that this kind of sequences satisfies a better estimate of discrepancy bound than the generalized Halton sequences.

Definition 1

Let \(a_1,\cdots , a_n\in {\mathbb {Z}}.\) We shall denote
$$\begin{aligned} (\overline{a_1,\cdots , a_n}) := (a_1,\cdots , a_n,a_1,\cdots , a_n,a_1,\cdots ,a_n,\cdots ) \end{aligned}$$
to be the periodic sequence of the integers \(a_1,\cdots , a_n.\) In addition, we shall sometimes abuse the following notation
$$\begin{aligned}&(a_1,\cdots ,a_n,a_{n+1},a_{n+2},a_{n+3},\cdots ) \\&\quad := (a_1,\cdots , a_n, a_{n+1\bmod {n}}, a_{n+2\bmod {n}},a_{n+3\bmod {n}},\cdots ) \end{aligned}$$
to mean the periodic sequence \((\overline{a_1,\cdots , a_n}),\) when it is clear from the context that \(a_{n+1},a_{n+2},a_{n+3},\cdots \) are not defined. Here, for each \(m\in {\mathbb {N}},\) \(m\bmod {n}\) denotes the remainder of the Euclidean division of m by n,  except \(m\bmod {n} = n\) when m is divisible by n.

Next we introduce the notion of admissible sequences of integers which extends Atanassov’s notion of admissible integers.

Definition 2

Let \(j_1,\cdots ,j_s\in {\mathbb {N}}.\) Suppose \(p_{1,1},\cdots , p_{1,j_1},\cdots , p_{s,1},\cdots , p_{s,j_s}\) are distinct prime numbers such that, for each \(1\le i\le s\), there exists a common primitive root modulo \(p_{i,1},\cdots , p_{i,j_i}.\) For each \(1\le i\le s,\) let \(p_i = (\overline{p_{i,1},\cdots , p_{i,j_i}})\). Periodic sequences of integers \(k_1 = (\overline{k_{1,1},\cdots , k_{1,j_1}}),\) \(\cdots ,\) \(k_s = (\overline{k_{s,1},\cdots , k_{s,j_s}})\) are said to be admissible for \(p_1,\cdots , p_s\) if, for each \((d_1,\cdots , d_s)\in {\mathbb {Z}}_{p_{1,\ell _1}}^* \times \cdots \times {\mathbb {Z}}_{p_{s,\ell _s}}^*\) (\(1\le \ell _i\le j_i,\) \(1\le i\le s\)), there exists \((\alpha _1,\cdots ,\alpha _s)\in {\mathbb {N}}^s\) such that
$$\begin{aligned} k_{i,1}\cdots k_{i,\alpha _i-1} \prod _{\begin{array}{c} 1\le i_0\le s \\ i_0\ne i \end{array}} p_{i_0,1}\cdots p_{i_0,\alpha _{i_0-1}} \equiv d_i \pmod {p_{i,\alpha _i}}\quad \text{ and }\quad \alpha _i\equiv \ell _i \pmod {j_i} \end{aligned}$$
for all \(1\le i\le s.\)

Note that the existence of admissible sequences for such prime sequences \(p_1,\cdots , p_s\) in Definition 2 will be proved in Lemma 6.

Definition 3

Let \(p_1= (\overline{p_{1,1},\cdots , p_{1,j_1}}),\) \(\cdots ,\) \(p_s = (\overline{p_{s,1},\cdots , p_{s,j_s}})\) be periodic sequences of distinct prime numbers such that, for each \(1\le i\le s\), there exists a common primitive root modulo \(p_{i,1},\cdots , p_{i,j_i}.\) Suppose \(k_1,\cdots , k_s\) are admissible sequences for \(p_1,\cdots , p_s.\) For each \(1\le i\le s,\) let \(\varSigma _i = (\sigma _{i,j})_{j=1}^\infty \) be the sequence of permutations of \({\mathbb {Z}}_{p_{i,1}}, {\mathbb {Z}}_{p_{i,2}},{\mathbb {Z}}_{p_{i,3}},\cdots \) such that
$$\begin{aligned} \sigma _{i,j}: {\mathbb {Z}}_{p_{i,j}}\rightarrow {\mathbb {Z}}_{p_{i,j}}: x\mapsto x k_{i,1}\cdots k_{i,j-1} \bmod {p_{i,j}}. \end{aligned}$$
The modified Halton sequence in Cantor bases \(p_1,\cdots , p_s\) with respect to \(k_1,\cdots , k_s\) is defined to be \((\phi _{p_1}^{\varSigma _1}(n),\cdots , \phi _{p_s}^{\varSigma _s}(n))_{n=0}^\infty .\)

When the sequences \(p_1,\cdots , p_s\) in Definition 3 are of period one, i.e. \(p_1=(\overline{p_{1,1}}),\cdots , p_s=(\overline{p_{s,1}}),\) our modified Halton sequence in Cantor bases is exactly the modified Halton sequence introduced by Atanassov [1].

The notion of admissible sequences seems technical and hard to understand, so it is worth noting here that this condition involves some periodic properties and is used to improve the estimate in (3) for \(\varLambda _1\). In particular, we shall be considering the distribution of the modified Halton sequence in Cantor bases over an elementary interval, which will be divided into \(\# ({\mathbb {Z}}_{p_{1,\alpha _1}}\times \cdots \times {\mathbb {Z}}_{p_{s,\alpha _s}})\) subintervals. The admissibility condition ensures that there will be the same number of elements of the sequence in each subinterval. These periodic properties will be seen in Lemma 8. Due to the fact that the subintervals of the considered elementary interval are small and that the exact number of elements of the sequence in each subinterval is known, it is possible to make a better estimate of the discrepancy bound for the modified Halton sequence in Cantor bases than for the generalized Halton sequence in Cantor bases.

The following statement is our second main result which gives an estimate of discrepancy bound of the modified Halton sequence in Cantor bases.

Theorem 2

Let \(\omega \) be the modified Halton sequence in Cantor bases \(p_1,\cdots , p_s\) with respect to \(k_1,\cdots , k_s.\) Then, for any \(N\in {\mathbb {N}},\) we have
$$\begin{aligned} D^*_N(\omega ) \le c \frac{(\log N)^s}{N} + O\left( \frac{(\log N)^{s-1}}{N}\right) \end{aligned}$$
with
$$\begin{aligned} c=c(p_1,\cdots , p_s) = \frac{1}{s!}\left( \sum _{i=1}^s\log M_i\right) \prod _{i=1}^s \frac{M_i(1+\log M_i)}{(m_i-1)\log m_i}, \end{aligned}$$
where \(M_i = \max (p_{i,1},\cdots , p_{i,j_i})\) and \(m_i = \min (p_{i,1},\cdots , p_{i,j_i})\) \((1\le i\le s).\)

Note that Theorem 2 gives a lower estimate \(c=c(p_1,\cdots , p_s)\) than the bound \(c=c(b_1,\cdots , b_s)\) provided by Theorem 1, when the \(m_i\)’s are large enough. Also, when the sequences \(p_1,\cdots , p_s\) are of period one, i.e. \(M_i=m_i\) for all \(1\le i\le s,\) the estimated bound \(c=c(p_1,\cdots , p_s)\) of our modified Halton sequence in Cantor bases is indeed the same as that of the modified Halton sequence given in [1]. Although the modified Halton sequences in Cantor bases do not attain a lower estimate of discrepancy bound than Atanassov’s modified Halton sequences, our method gives more variety of sequences with similar estimated bound, especially when \(M_i\) is large and when the difference \(M_i-m_i\) is small compared with \(M_i\) for each \(1\le i <s.\) This follows from the same argument as that at the end of Sect. 2.

4 Proof of Theorem 1

The proof of Theorem 1 is indeed inspired by and closely related to that given by Atanassov [1]. Moreover, it can be seen as an extension of that given by Haddley et al. [10]. Note that Lemma 1 is required to make the extension of the proof provided by Haddley et al. [10] possible.

In order to prove the theorem, we need the following five lemmas.

The first preliminary result is a variant of the Chinese remainder theorem, and it is used to prove Lemma 2.

Lemma 1

Let \(b_1 = (b_{1,j})_{j=1}^\infty , \cdots , b_s = (b_{s,j})_{j=1}^\infty \) be s arbitrary sequences of natural numbers greater than 1 such that, for all \(1\le i_1 < i_2 \le s\) and all \(j_1,j_2\in {\mathbb {N}},\) \(b_{i_1,j_1}\) and \(b_{i_2,j_2}\) are coprime. For each \(1\le i\le s\) and \(j\in {\mathbb {N}},\) let \(\sigma _{i,j}\) be a permutation of \({\mathbb {Z}}_{b_{i,j}},\) and let \(f_i:{\mathbb {N}}_0\rightarrow {\mathbb {N}}_0\) be a function defined by
$$\begin{aligned} f_i(n) = f_i\left( \sum _{j=1}^\infty n_{i,j}b_{i,1}\cdots b_{i,j-1} \right) = \sum _{j=1}^\infty \sigma _{i,j}(n_{i,j}) b_{i,1}\cdots b_{i,j-1}, \end{aligned}$$
for every \(n\in {\mathbb {N}}_0\) with the \(b_i\)-adic expansion \(\sum _{j=1}^\infty n_{i,j}b_{i,1}\cdots b_{i,j-1}.\) For each \(1\le i\le s,\) let \(\alpha _i\) be a natural number, and let \(l_{i,1}\in {\mathbb {Z}}_{b_{i,1}},\cdots , l_{i,\alpha _i}\in {\mathbb {Z}}_{b_{i,\alpha _i}}.\) Then there exists a unique \(0\le n < \prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i}\) such that, for all \(1\le i\le s,\)
$$\begin{aligned} f_i(n) \equiv l_{i,1} + l_{i,2}b_{i,1} + l_{i,3}b_{i,1}b_{i,2} + \cdots l_{i,\alpha _i} b_{i,1}\cdots b_{i,\alpha _i-1} \pmod {b_{i,1}\cdots b_{i,\alpha _i}}. \end{aligned}$$

Proof

For each \(1\le i\le s,\) let \(b^*_i = b_{i,1}\cdots b_{i,\alpha _i}.\) We first prove the uniqueness of the n. Suppose that n and \(n'\) are two solutions of all the congruences such that \(0\le n, n' < \prod _{i=1}^s b^*_i.\) It follows that \(f_i(n) \equiv f_i(n') \pmod {b^*_i}\) for all \(1\le i\le s,\) that is, we have
$$\begin{aligned} \sum _{j=1}^\infty \sigma _{i,j}(n_{i,j}) b_{i,1}\cdots b_{i,j-1} \equiv \sum _{j=1}^\infty \sigma _{i,j}(n'_{i,j}) b_{i,1}\cdots b_{i,j-1} \pmod {b^*_i}, \end{aligned}$$
and this is equivalent to
$$\begin{aligned} \sum _{j=1}^{\alpha _i} \sigma _{i,j}(n_{i,j}) b_{i,1}\cdots b_{i,j-1} \equiv \sum _{j=1}^{\alpha _i} \sigma _{i,j}(n'_{i,j}) b_{i,1}\cdots b_{i,j-1} \pmod {b^*_i}. \end{aligned}$$
We know that \(|\sum _{j=1}^{\alpha _i} \sigma _{i,j}(n_{i,j}) b_{i,1}\cdots b_{i,j-1} - \sum _{j=1}^{\alpha _i} \sigma _{i,j}(n'_{i,j}) b_{i,1}\cdots b_{i,j-1}| < b^*_i\) for each \(1\le i\le s,\) and hence this difference must be 0. Therefore, we obtain \(\sigma _{i,j}(n_{i,j}) = \sigma _{i,j}(n'_{i,j})\) for all \(1\le i\le s\) and \(1\le j\le \alpha _i.\) Since \(\sigma _{i,j}\) are all bijective, we have \(n_{i,j} = n'_{i,j}\) for all such i and j. It follows that \(b^*_i\mid n-n'\) for each i. This implies that \(b^*_1\cdots b^*_s \mid n-n'\) because \(b^*_1,\cdots , b^*_s\) are pairwise coprime. By the choice of n and \(n',\) we must have \(n=n'.\)
Now we show the existence of such n. Define \(F:{\mathbb {Z}}_{b^*_1\cdots b^*_s} \rightarrow {\mathbb {Z}}_{b^*_1}\times \cdots \times {\mathbb {Z}}_{b^*_s}\) by
$$\begin{aligned} F(n) = (f_1(n) \bmod {b^*_1},\cdots , f_s(n) \bmod {b^*_s}). \end{aligned}$$
It suffices to show that F is a bijection. By the proof of uniqueness, F must be an injection. For each \(1\le i\le s,\) it is clear that \(f_i\) is a bijection on \({\mathbb {Z}}_{b^*_i}.\) It follows immediately that F is a surjection since the domain and the codomain of F have the same number of elements. This proves the existence of such n. \(\square \)

The following lemma is a consequence of the so-called “elementary interval property” satisfied by Halton sequences.

Lemma 2

Let \(b_1 = (b_{1,j})_{j=1}^\infty , \cdots , b_s = (b_{s,j})_{j=1}^\infty \) be s arbitrary sequences of natural numbers greater than 1 such that, for all \(1\le i_1 < i_2 \le s\) and all \(j_1,j_2\in {\mathbb {N}},\) \(b_{i_1,j_1}\) and \(b_{i_2,j_2}\) are coprime. For each \(1\le i\le s\) and \(j\in {\mathbb {N}},\) let \(\sigma _{i,j}\) be a permutation of \({\mathbb {Z}}_{b_{i,j}}.\) For each \(1\le i\le s,\) denote \(\varSigma _i = (\sigma _{i,j})_{j=1}^\infty .\) Suppose that \(\omega \) is the generalized Halton sequence in Cantor bases \(b_1,\cdots ,b_s\) with respect to \(\varSigma _1,\cdots ,\varSigma _s.\) Let J be an interval of the form
$$\begin{aligned} J = \prod _{i=1}^s\left[ \frac{u_i}{b_{i,1}\cdots b_{i,\alpha _i}}, \frac{v_i}{b_{i,1}\cdots b_{i,\alpha _i}}\right) \end{aligned}$$
with integers \(0\le u_i <v_i \le b_{i,1}\cdots b_{i,\alpha _i}\) and \(\alpha _i\in {\mathbb {N}}\) \((1\le i\le s).\) Then
$$\begin{aligned} |A(J;N;\omega ) - N\lambda _s(J)| \le \prod _{i=1}^s (v_i-u_i) \end{aligned}$$
holds for every \(N\in {\mathbb {N}}.\) Moreover, for every \(N\le \prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i},\) we have \(A(J;N;\omega ) \le \prod _{i=1}^s (v_i-u_i).\)

Proof

For each \(n\in {\mathbb {N}}_0,\) we denote the \(b_i\)-adic expansion of n by
$$\begin{aligned} n = \sum _{j=1}^\infty n_{i,j}b_{i,1}\cdots b_{i,j-1} = n_{i,1} + n_{i,2}b_{i,1} + n_{i,3}b_{i,1}b_{i,2} + n_{i,4}b_{i,1}b_{i,2}b_{i,3} + \cdots , \end{aligned}$$
where \(n_{i,j} \in {\mathbb {Z}}_{b_{i,j}}\) \((j\in {\mathbb {N}}).\) Let \(\ell = (l_1,\cdots , l_s)\in {\mathbb {N}}_0^s\) be such that, for all \(1\le i\le s,\) we have \(0\le l_i <b_{i,1}\cdots b_{i,\alpha _i}\) with the expansion
$$\begin{aligned} l_i = l_{i,\alpha _i} + l_{i,\alpha _i-1}b_{i,\alpha _i} + l_{i,\alpha _i-2}b_{i,\alpha _i}b_{i,\alpha _i-1} + \cdots + l_{i,1}b_{i,\alpha _i}\cdots b_{i,2}, \end{aligned}$$
where \(l_{i,\alpha _i-j}\in {\mathbb {Z}}_{b_{i,\alpha _i-j}}\) \((0\le j\le \alpha _i-1).\) We consider the interval
$$\begin{aligned} J_{\ell } = \prod _{i=1}^s\left[ \frac{l_i}{b_{i,1}\cdots b_{i,\alpha _i}}, \frac{l_i+1}{b_{i,1}\cdots b_{i,\alpha _i}} \right) . \end{aligned}$$
Then the nth element \(\omega _n\) of the generalized Halton sequence in Cantor bases is contained in \(J_\ell \) if and only if, for all \(1\le i\le s,\)
$$\begin{aligned}&\frac{l_{i,1}}{b_{i,1}} + \cdots + \frac{l_{i,\alpha _i}}{b_{i,1}\cdots b_{i,\alpha _i}} \le \frac{\sigma _{i,1}(n_{i,1})}{b_{i,1}} + \frac{\sigma _{i,2}(n_{i,2})}{b_{i,1}b_{i,2}} + \cdots \\&\quad < \frac{l_{i,1}}{b_{i,1}} + \cdots + \frac{l_{i,\alpha _i}}{b_{i,1}\cdots b_{i,\alpha _i}} + \frac{1}{b_{i,1}\cdots b_{i,\alpha _i}}. \end{aligned}$$
This is however equivalent to \(\sigma _{i,1}(n_{i,1}) = l_{i,1}, \cdots , \sigma _{i,\alpha _i}(n_{i,\alpha _i}) = l_{i,\alpha _i}\) which in turn is equivalent to
$$\begin{aligned}&\sum _{j=1}^\infty \sigma _{i,j}(n_{i,j}) b_{i,1}\cdots b_{i,j-1} \equiv l_{i,1} + l_{i,2}b_{i,1} + \cdots + l_{i,\alpha _i} b_{i,1}\cdots b_{i,\alpha _i-1}\\&\quad \pmod {b_{i,1}\cdots b_{i,\alpha _i}} \end{aligned}$$
for all \(1\le i\le s.\) By Lemma 1, every \(\prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i}\) consecutive elements of the generalized Halton sequence in Cantor bases contain exactly one element in \(J_\ell ,\) in other words, \(A(J_\ell ; t\prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i} ; \omega ) = t\) for all \(t\in {\mathbb {N}},\) and hence
$$\begin{aligned} A\left( J_\ell ; t\prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i}; \omega \right) - \left( t\prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i}\right) \lambda _s(J_\ell ) = 0. \end{aligned}$$
Therefore, for every \(N\in {\mathbb {N}},\) we obtain
$$\begin{aligned} \left| A(J_\ell ; N;\omega ) - N\lambda _s(J_\ell ) \right| \le 1. \end{aligned}$$
Now we write the interval J as a disjoint union of intervals of the form \(J_\ell ,\)
$$\begin{aligned} J = \bigcup _{l_1 = u_1}^{v_1 - 1} \cdots \bigcup _{l_s = u_s}^{v_s-1} J_\ell , \end{aligned}$$
where \(\ell = (l_1,\cdots , l_s).\) We then have
$$\begin{aligned} |A(J;N;\omega ) - N\lambda _s(J)| \le \sum _{l_1=u_1}^{v_1-1}\cdots \sum _{l_s=u_s}^{v_s-1} |A(J_\ell ; N;\omega ) - N\lambda _s(J_\ell )| \le \prod _{i=1}^s (v_i-u_i), \end{aligned}$$
which proves the first assertion.
For every \(N\le \prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i},\) we always have \(A(J_\ell ; N;\omega ) \le 1\) for each \(\ell = (l_1,\cdots , l_s)\in {\mathbb {N}}_0^s\) with \(0\le l_i < b_{i,1}\cdots b_{i,\alpha _i}\) for all \(1\le i\le s,\) and hence
$$\begin{aligned} A(J;N;\omega ) = \sum _{l_1=u_1}^{v_1-1}\cdots \sum _{l_s=u_s}^{v_s-1} A(J_\ell ; N;\omega ) \le \prod _{i=1}^s (v_i-u_i). \end{aligned}$$
This is the second assertion of the lemma. \(\square \)

The following lemma, which is borrowed from [10], is important for achieving an s! factor in the bounds for the discrepancy.

Lemma 3

[10, Lemma 3.3] Let \(b_1 = (b_{1,j})_{j=1}^\infty , \cdots , b_s = (b_{s,j})_{j=1}^\infty \) be s arbitrary sequences of natural numbers greater than 1. Suppose \((a_{1,\alpha })_{\alpha =0}^\infty ,\cdots , (a_{s,\alpha })_{\alpha =0}^\infty \) are s bounded sequences of nonnegative real numbers such that \(a_{i,0} \le 1\) and \(a_{i,\alpha } \le f_i\) for some fixed \(f_i>0\) and for each \(\alpha \in {\mathbb {N}}\) and \(1\le i\le s.\) Then, for any \(N\in {\mathbb {N}},\) we have
$$\begin{aligned} \sum _{\begin{array}{c} (\alpha _1,\cdots ,\alpha _s)\in {\mathbb {N}}_0^s \\ \prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i} \le N \end{array}} \prod _{i=1}^s a_{i,\alpha _i} \le \frac{1}{s!}\prod _{i=1}^s \left( f_i\frac{\log N}{\log m_i} + s\right) , \end{aligned}$$
where \(m_i = \min (b_{i,j})_{j=1}^\infty \) \((1\le i\le s).\)

The proof of this lemma is based on an argument of Diophantine geometry which asserts that the number of positive solutions \((\alpha _1,\cdots , \alpha _s)\) of the inequality \(\prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i} \le N\) is bounded by \(\frac{1}{s!}\prod _{i=1}^s \frac{\log N}{\log m_i}.\)

Next we need to introduce some notation. Let \(J\subseteq \mathbb {R}^s\) be an interval. Then a signed splitting of J is a collection of not necessarily disjoint intervals \(J_1,\cdots , J_r\) together with signs \(\varepsilon _1,\cdots ,\varepsilon _r\in \{-1,1\}\) such that, for all \(x\in J,\) we have
$$\begin{aligned} \sum _{\begin{array}{c} i=1 \\ x\in J_i \end{array}}^r \varepsilon _i = 1. \end{aligned}$$
A function \(\nu \) on the class of intervals in \(\mathbb {R}^s\) is said to be additive if, for each pair of disjoint intervals AB in \(\mathbb {R}^s,\) we have \(\nu (A\cup B)=\nu (A)+\nu (B).\) It is not hard to see that the s-dimensional Lebesgue measure \(\lambda _s\) and the counting function \(A(\cdot ;N;\omega )\) are the examples we are particularly interested in. It is not hard to check that, for any additive function \(\nu \) on the class of intervals in \(\mathbb {R}^s,\) we have
$$\begin{aligned} \nu (J) = \sum _{i=1}^r\varepsilon _i\nu (J_i\cap J), \end{aligned}$$
where \((J_1,\cdots ,J_r;\varepsilon _1,\cdots ,\varepsilon _r)\) is a signed splitting of J. The following lemma is borrowed from [1] (see also [5] for a detailed proof).

Lemma 4

[1, Lemma 3.5] Let \(J = \prod _{i=1}^s [0,z_i)\) be an s-dimensional interval. For each \(1\le i\le s,\) let \((z_{i,\alpha })_{\alpha = 1,\cdots , n_i}\) be an arbitrary finite sequence of numbers in [0, 1]. Define further \(z_{i,0} = 0\) and \(z_{i,n_i+1} = z_i\) for all \(1\le i\le s.\) Then the collection of intervals
$$\begin{aligned} \prod _{i=1}^s [\min (z_{i,\alpha _i}, z_{i,\alpha _i+1}), \max (z_{i,\alpha _i}, z_{i,\alpha _i+1})) \end{aligned}$$
together with the signs \(\varepsilon _{\alpha _1,\cdots , \alpha _s} =\prod _{i=1}^s \mathrm {sgn}(z_{i,\alpha _i+1} - z_{i,\alpha _i}),\) for \(0\le \alpha _i \le n_i\) and \(1\le i\le s,\) defines a signed splitting of the interval J.

The signed splitting technique is interesting here because it will lead to the improvement by a \(2^s\) factor in the bounds for the discrepancy. In order to use it, we need a digit expansion of reals \(z\in [0,1)\) in \((b_j)_{j=1}^\infty \)-adic base which uses signed digits. The next lemma, from [10], shows that such an expansion exists. Note that signed splittings coupled with signed numeration systems were first introduced in [7, 9].

Lemma 5

[10, Lemma 3.5] Let \(b = (b_j)_{j=1}^\infty \) be an arbitrary sequence of natural numbers greater than 1. Then every \(z\in [0,1)\) can be written in the form
$$\begin{aligned} z = a_0 + \frac{a_1}{b_1} + \frac{a_2}{b_1b_2} + \frac{a_3}{b_1b_2b_3} + \cdots \end{aligned}$$
with integer digits \(a_0,a_1,a_2,\cdots \) such that \(a_0 \in \{0,1\}\) and \(-\lfloor \frac{b_j-1}{2}\rfloor \le a_j \le \lfloor \frac{b_j}{2}\rfloor \) for all \(j\in {\mathbb {N}}.\) This expansion is called the signed b-adic expansion of z.

Now we are ready to prove our first main theorem.

Proof (Proof of Theorem 1)     Let \(J= \prod _{i=1}^s [0,z_i) \subseteq [0,1)^s.\) According to Lemma 5, for all \(1\le i\le s,\) we consider the signed \(b_i\)-adic expansion of \(z_i\) of the form
$$\begin{aligned} z_i = a_{i,0} + \frac{a_{i,1}}{b_{i,1}} + \frac{a_{i,2}}{b_{i,1}b_{i,2}} + \frac{a_{i,3}}{b_{i,1}b_{i,2}b_{i,3}} + \cdots \end{aligned}$$
with \(a_{i,0}\in \{0,1\}\) and \(-\lfloor \frac{b_{i,j}-1}{2}\rfloor \le a_{i,j} \le \lfloor \frac{b_{i,j}}{2}\rfloor \) \((j\in {\mathbb {N}}).\)
For each \(1\le i\le s,\) let \(n_i = \lfloor \frac{\log N}{\log m_i}\rfloor + 1,\) and, for each \(1\le \alpha \le n_i,\) define the truncation of the expansion
$$\begin{aligned} z_{i,\alpha } = a_{i,0} + \frac{a_{i,1}}{b_{i,1}} + \frac{a_{i,2}}{b_{i,1}b_{i,2}} + \cdots + \frac{a_{i,\alpha -1}}{b_{i,1}\cdots b_{i,\alpha -1}}, \end{aligned}$$
and let \(z_{i,0} = 0\) and \(z_{i,n_i+1} = z_i.\)
By Lemma 4, the collection of intervals
$$\begin{aligned} J_{\alpha _1,\cdots ,\alpha _s} = \prod _{i=1}^s [\min (z_{i,\alpha _i}, z_{i,\alpha _i+1}), \max (z_{i,\alpha _i}, z_{i,\alpha _i+1})) \end{aligned}$$
together with the signs \(\varepsilon _{\alpha _1,\cdots ,\alpha _s} = \prod _{i=1}^s \mathrm {sgn}(z_{i,\alpha _i+1}-z_{i,\alpha _i}),\) for \(0\le \alpha _i\le n_i\) and \(1\le i\le s,\) defines a signed splitting of the interval J.
Since both \(\lambda _s\) and \(A(\cdot ; N;\omega )\) are additive functions on the set of intervals, we obtain that
$$\begin{aligned} \begin{aligned} A(J;N;\omega )-N\lambda _s(J)&= \sum _{\alpha _1=0}^{n_1}\cdots \sum _{\alpha _s=0}^{n_s} \varepsilon _{\alpha _1,\cdots , \alpha _s}( A(J_{\alpha _1,\cdots , \alpha _s};N;\omega ) - N\lambda _s(J_{\alpha _1,\cdots , \alpha _s}) ) \\&= \varLambda _1 + \varLambda _2, \end{aligned} \end{aligned}$$
(3)
where \(\varLambda _1\) denotes the sum over all \((\alpha _1,\cdots , \alpha _s)\) such that \(\prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i} \le N\) and \(\varLambda _2\) denotes the remaining part of the above sum.
First we deal with the sum \(\varLambda _1.\) For each \(1\le i\le s,\) the length of the interval \([\min (z_{i,\alpha _i}, z_{i,\alpha _i+1}), \max (z_{i,\alpha _i}, z_{i,\alpha _i+1}))\) is \(|a_{i,\alpha _i} / b_{i,1}\cdots b_{i,\alpha _i}|,\) and the endpoints of this interval are rationals with denominator \(b_{i,1}\cdots b_{i,\alpha _i}.\) It is worth noting that, due to the choice of \(n_i,\) we have \(\alpha _i <n_i\) when \(\prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i} \le N.\) Accordingly, the intervals \(J_{\alpha _1,\cdots ,\alpha _s}\) are of the form as considered in Lemma 2 from which we obtain
$$\begin{aligned} |A(J_{\alpha _1,\cdots , \alpha _s}; N;\omega ) - N\lambda _s(J_{\alpha _1,\cdots ,\alpha _s})| \le \prod _{i=1}^s |a_{i,\alpha _i}|. \end{aligned}$$
We have \(|a_{i,\alpha _i}|\le \lfloor b_{i,\alpha _i}/2\rfloor \le \lfloor M_i/2\rfloor =: f_i.\) An application of Lemma 3 yields that
$$\begin{aligned} \varLambda _1 \le \frac{1}{s!}\prod _{i=1}^s\bigg ( \frac{\lfloor M_i/2\rfloor \log N}{\log m_i} + s \bigg ). \end{aligned}$$
It remains to estimate \(\varLambda _2.\) To this end, we split the set of s-tuples \((\alpha _1,\cdots , \alpha _s)\) for which \(\prod _{i=1}^s b_{i,1}\cdots b_{i,\alpha _i} >N\) into disjoint sets \(B_0, B_1,\cdots , B_{s-1}\) where we set \(B_0 = \{(\alpha _1,\cdots , \alpha _s)\in {\mathbb {N}}_0^s:b_{1,1}\cdots b_{1,\alpha _1} >N\}\) and, for \(1\le l\le s-1,\)
$$\begin{aligned} B_l = \left\{ (\alpha _1,\cdots , \alpha _s)\in {\mathbb {N}}_0^s :\prod _{i=1}^l b_{i,1}\cdots b_{i,\alpha _i} \le N\>\text{ and }\> \prod _{i=1}^{l+1} b_{i,1}\cdots b_{i,\alpha _i} >N \right\} . \end{aligned}$$
Here we abuse the notation \({\mathbb {N}}_0^s\). The choices of \((\alpha _1,\cdots ,\alpha _s)\) must also satisfy \(\alpha _i\le n_i\) for each \(1\le i\le s.\)
For a fixed \(1\le l\le s-1\) and a fixed l-tuple \((\alpha _1,\cdots , \alpha _l)\) with \(\prod _{i=1}^l b_{i,1}\cdots b_{i,\alpha _i} \le N,\) define r to be the largest integers such that
$$\begin{aligned} \left( \prod _{i=1}^l b_{i,1}\cdots b_{i,\alpha _i}\right) (b_{l+1, 1}\cdots b_{l+1, r-1} )\le N. \end{aligned}$$
It follows that the tuple \((\alpha _1,\cdots , \alpha _l,\alpha _{l+1},\cdots , \alpha _s)\) is contained in \(B_l\) if and only if \(\alpha _{l+1}\ge r.\)
Therefore, for each \(0\le l\le s-1\) and fixed \(\alpha _1,\cdots , \alpha _l\in {\mathbb {N}}_0\) such that \(\prod _{i=1}^l b_{i,1}\cdots b_{i,\alpha _i} \le N,\) we have
$$\begin{aligned}&\sum _{\begin{array}{c} \alpha _{l+1},\cdots , \alpha _s\in {\mathbb {N}}_0 \\ (\alpha _1,\cdots ,\alpha _l,\alpha _{l+1},\cdots ,\alpha _s)\in B_l \end{array}} \varepsilon _{\alpha _1,\cdots , \alpha _s}( A(J_{\alpha _1,\cdots , \alpha _s};N;\omega ) - N\lambda _s(J_{\alpha _1,\cdots , \alpha _s}))\\&\quad = \pm \, (A(L;N;\omega )-N\lambda _s(L)), \end{aligned}$$
where
$$\begin{aligned} L= & {} \prod _{i=1}^l [\min (z_{i,\alpha _i},z_{i,\alpha _i+1}),\max (z_{i,\alpha _i}, z_{i,\alpha _i+1})) \\&\quad \times [\min (z_{l+1,r},z_{l+1}),\max (z_{l+1,r},z_{l+1}))\times \prod _{i=l+2}^s [0,z_i). \end{aligned}$$
Let \((\alpha _1,\cdots , \alpha _s)\in B_l.\) Since we have
$$\begin{aligned} {\begin{aligned}&|z_{l+1}-z_{l+1,r}| = \left| \frac{a_{l+1,r}}{b_{l+1,1}\cdots b_{l+1,r}} +\frac{a_{l+1,r+1}}{b_{l+1,1}\cdots b_{l+1,r+1}} + \frac{a_{l+1,r+2}}{b_{l+1,1}\cdots b_{l+1,r+2}} +\cdots \right| \\&\quad = \frac{1}{b_{l+1,1}\cdots b_{l+1,r-1}}\left| \frac{a_{l+1,r}}{b_{l+1,r}} +\frac{a_{l+1,r+1}}{b_{l+1,r} b_{l+1,r+1}} + \frac{a_{l+1,r+2}}{b_{l+1,r}b_{l+1,r+1}b_{l+1,r+2}} +\cdots \right| \\&\quad \le \frac{1}{b_{l+1,1}\cdots b_{l+1,r-1}}\left( \frac{\lfloor b_{l+1,r}/2\rfloor }{b_{l+1,r}}+\frac{\lfloor b_{l+1,r+1}/2\rfloor }{b_{l+1,r} b_{l+1,r+1}}\right. \\&\qquad \left. +\, \frac{\lfloor b_{l+1,r+2}/2 \rfloor }{b_{l+1,r}b_{l+1,r+1}b_{l+1,r+2}} +\cdots \right) \\&\quad \le \frac{1}{b_{l+1,1}\cdots b_{l+1,r-1}}\left( \frac{1}{2}+\frac{1}{2 b_{l+1,r}} + \frac{1}{2b_{l+1,r}b_{l+1,r+1}}+\cdots \right) \\&\quad \le \frac{1}{b_{l+1,1}\cdots b_{l+1,r-1}}\left( \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3}+\cdots \right) \\&\quad = \frac{1}{b_{l+1,1}\cdots b_{l+1,r-1}}, \end{aligned}} \end{aligned}$$
the interval \([\min (z_{l+1,r},z_{l+1}),\max (z_{l+1,r},z_{l+1}))\) is contained in some interval
$$\begin{aligned} \left[ \frac{u}{b_{l+1,1}\cdots b_{l+1,r}}, \frac{v}{b_{l+1,1}\cdots b_{l+1,r}}\right) \end{aligned}$$
for \(u,v\in {\mathbb {N}}_0\) with \(v-u \le b_{l+1,r}.\) Hence, L is contained in the interval
$$\begin{aligned} L'= & {} \prod _{i=1}^l [\min (z_{i,\alpha _i},z_{i,\alpha _i+1}),\max (z_{i,\alpha _i}, z_{i,\alpha _i+1}))\\&\times \left[ \frac{u}{b_{l+1,1}\cdots b_{l+1,r}}, \frac{v}{b_{l+1,1}\cdots b_{l+1,r}}\right) \times [0,1)^{s-l-1}. \end{aligned}$$
Since \((\alpha _1,\cdots ,\alpha _s)\in B_l,\) we have \((\prod _{i=1}^l b_{i,1}\cdots b_{i,\alpha _i})(b_{l+1,1}\cdots b_{l+1,r}) >N\) and \(\prod _{i=1}^l b_{i,1}\cdots b_{i,\alpha _i} \le N.\) The latter inequality implies that \(\alpha _i <n_i\) for every \(1\le i\le l.\) Thus, an application of Lemma 2 yields that
$$\begin{aligned} A(L;N;\omega ) \le A(L';N;\omega ) \le b_{l+1,r} \prod _{i=1}^l |a_{i,\alpha _i}|. \end{aligned}$$
But on the other hand, we also have \(N\lambda _s(L) \le b_{l+1,r}\prod _{i=1}^l |a_{i,\alpha _i}|.\) Hence,
$$\begin{aligned} |A(L;N;\omega ) - N\lambda _s(L)| \le b_{l+1,r}\prod _{i=1}^l |a_{i,\alpha _i}| \le M_{l+1} \prod _{i=1}^l c_{i,\alpha _i}, \end{aligned}$$
where \(c_{i,\alpha _i} =1 \) if \(\alpha _i=0\) and \(c_{i,\alpha _i} = \lfloor M_i/2 \rfloor \) otherwise.
Summing up, we obtain
$$\begin{aligned} \begin{aligned} |\varLambda _2|&\le \sum _{l=0}^{s-1} \sum _{\begin{array}{c} \alpha _1,\cdots , \alpha _l\in {\mathbb {N}}_0 \\ \prod _{i=1}^l b_{i,1}\cdots b_{i,\alpha _i} \le N \end{array}} \left| \sum _{(\alpha _1,\cdots , \alpha _s)\in B_l} \epsilon _{\alpha _1,\cdots , \alpha _s}(A(J_{\alpha _1,\cdots , \alpha _s};N;\omega ) \right. \\&\quad \left. - N\lambda _s(J_{\alpha _1,\cdots , \alpha _s}) ) \right| \\&\le \sum _{l=0}^{s-1} \sum _{\begin{array}{c} \alpha _1,\cdots , \alpha _l\in {\mathbb {N}}_0 \\ \prod _{i=1}^l b_{i,1}\cdots b_{i,\alpha _i} \le N \end{array}} M_{l+1} \prod _{i=1}^l c_{i,\alpha _i} \\&\le \sum _{l=0}^{s-1}\frac{M_{l+1}}{l!}\prod _{i=1}^l\left( \frac{\lfloor M_i/2\rfloor \log N}{\log m_i} +l \right) , \end{aligned} \end{aligned}$$
(4)
where we have used Lemma 3 again. Hence, the result follows. \(\square \)

5 Proof of Theorem 2

Lemma 6

Suppose that \(p_1= (\overline{p_{1,1},\cdots , p_{1,j_1}}),\) \(\cdots ,\) \(p_s = (\overline{p_{s,1},\cdots , p_{s,j_s}})\) are periodic sequences of distinct prime numbers such that, for each \(1\le i\le s\), there exists a common primitive root modulo \(p_{i,1},\cdots , p_{i,j_i}.\) Then there exist admissible sequences \(k_1 = (\overline{k_{1,1},\cdots , k_{1,j_1}}),\) \(\cdots ,\) \(k_s = (\overline{k_{s,1},\cdots , k_{s,j_s}})\) for \(p_1,\cdots , p_s.\)

Proof

For each \(1\le i\le s,\) let \(g_i\) be some fixed common primitive root modulo \(p_{i,1}, \cdots , p_{i,j_i}.\) The congruences in Definition 2 lead to the system \((1\le i\le s)\)
$$\begin{aligned} \begin{aligned}&g_i^{(a_{i,i,1} + \cdots +a_{i,i,j_i})x_i +a_{i,i,1}+\cdots +a_{i,i,\ell _i-1} + \sum _{i_0\ne i}( (a_{i,i_0,1} +\cdots + a_{i,i_0,j_{i_0}})x_{i_0} + a_{i,i_0,1}+\cdots + a_{i,i_0,\ell _{i_0}-1} )} \\&\quad \equiv g_i^{c_i} \pmod {p_{i,\ell _i}}, \quad i = 1,\cdots , s, \end{aligned} \end{aligned}$$
(5)
where \(g_i^{a_{i,i_0,1}} \equiv p_{i_0,1}\pmod {p_{i,\ell _i}},\) \(\cdots ,\) \(g_i^{a_{i,i_0,j_{i_0}}} \equiv p_{i_0,j_{i_0}}\pmod {p_{i,\ell _i}}\) for \(i_0\ne i,\) \(g_i^{a_{i,i,1}} \equiv k_{i,1} \pmod {p_{i,\ell _i}},\) \(\cdots ,\) \(g_i^{a_{i,i,j_i}} \equiv k_{i,j_i} \pmod {p_{i,\ell _i}},\) \(g_i^{c_i} \equiv b_i\pmod {p_{i,\ell _i}},\) and \(\alpha _i = j_i x_i +\ell _i.\) It is worth noting that the choice of each integer \(k_{i,j}\) can be fixed according to the Chinese remainder theorem and the fact that \(p_{i,1}, \cdots , p_{i,j_i}\) are all distinct primes. The system of congruences in (5) is equivalent to
$$\begin{aligned} \begin{aligned}&(a_{i,i,1} + \cdots +a_{i,i,j_i})x_i +a_{i,i,1}+\cdots +a_{i,i,\ell _i-1}\\&\qquad + \sum _{i_0\ne i}( a_{i,i_0}x_{i_0} + a_{i,i_0,1}+\cdots + a_{i,i_0,\ell _{i_0}-1} ) \\&\quad \equiv c_i \pmod {p_{i,\ell _i} -1}, \quad i=1,\cdots , s, \end{aligned} \end{aligned}$$
(6)
where \(a_{i,i_0} = a_{i,i_0,1}+\cdots + a_{i,i_0,j_{i_0}}\) for \(i_0\ne i.\) We introduce s integer variables \(y_1,\cdots , y_s\) to change the congruences (6) into a system of Diophantine equations
$$\begin{aligned} (a_{i,i,1} + \cdots +a_{i,i,j_i})x_i + \sum _{i_0\ne i}a_{i,i_0}x_{i_0} = c_i' + y_i(p_{i,\ell _i}-1), \quad i=1,\cdots , s, \end{aligned}$$
(7)
where \(c_i' = c_i - \sum _{i_0=1}^s (a_{i,i_0,1}+\cdots + a_{i,i_0,\ell _{i_0-1}}).\)
In order to prove the lemma, it suffices to show, for any given integers \(a_{i,i_0}\) with \(i\ne i_0,\) the existence of integers \(a_{1,1,1}, \cdots , a_{1,1,j_1},\cdots , a_{s,s,1},\cdots , a_{s,s,j_s}\) such that, for any integers \(c_1',\cdots , c_s'\) and any integers \(y_1,\cdots , y_s\), the system (7) has a solution in integers \(x_1,\cdots , x_s.\) Note that we actually require \(x_i\in {\mathbb {N}}_0\) so that \(\alpha _i = j_ix_i +\ell _i\in {\mathbb {N}}_0,\) but this nonnegativity of \(x_i\) can be achieved by a suitable choice of \(y_1,\cdots , y_s.\) Let
$$\begin{aligned} A = \begin{pmatrix} a_{1,1,1}+\cdots + a_{1,1,j_1} &{}\quad a_{1,2} &{}\quad \cdots &{}\quad a_{1,s} \\ a_{2,1} &{}\quad a_{2,2,1}+\cdots + a_{2,2,j_2} &{}\quad \cdots &{}\quad a_{2,s} \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ a_{s,1} &{}\quad a_{s,2} &{}\quad \cdots &{}\quad a_{s,s,1}+\cdots + a_{s,s,j_s} \end{pmatrix}. \end{aligned}$$
By Cramer’s Rule, it is enough to show that the determinant of A can be made to be 1 by a suitable choice of the numbers \(a_{1,1,1}, \cdots , a_{1,1,j_1},\cdots , a_{s,s,1},\cdots , a_{s,s,j_s}\). This claim follows by induction on s. When \(s=1,\) choose \(a_{1,1,1} = 1\) and \(a_{1,1,2} = \cdots = a_{1,1,j_1} = 0.\) Next, expand the determinant of A along the last column, \(A_{i,s}\) being the cofactors:
$$\begin{aligned} \det (A) = a_{1,s}A_{1,s} + a_{2,s}A_{2,s} +\cdots + a_{s-1,s} A_{s-1,s} + (a_{s,s,1}+\cdots +a_{s,s,j_s}) A_{s,s}. \end{aligned}$$
By the induction hypothesis, we have \(A_{s,s} = 1.\) Setting
$$\begin{aligned} a_{s,s,1}+\cdots +a_{s,s,j_s} = 1 - (a_{1,s}A_{1,s} + a_{2,s}A_{2,s} +\cdots + a_{s-1,s} A_{s-1,s}) \end{aligned}$$
yields \(\det (A) = 1,\) with an appropriate choice of integers \(a_{s,s,1},\cdots , a_{s,s,j_s}.\) This proves the existence of admissible sequences \(k_1,\cdots ,k_s.\) \(\square \)
Before proceeding, we introduce here some notations for brevity. Suppose \(p_1= (\overline{p_{1,1},\cdots , p_{1,j_1}}),\) \(\cdots ,\) \(p_s = (\overline{p_{s,1},\cdots , p_{s,j_s}})\) are periodic sequences of distinct prime numbers such that, for each \(1\le i\le s\), there exists a common primitive root modulo \(p_{i,1},\cdots , p_{i,j_i}.\) Let \(k_1 = (\overline{k_{1,1},\cdots , k_{1,j_1}}),\) \(\cdots ,\) \(k_s = (\overline{k_{s,1},\cdots , k_{s,j_s}})\) be admissible sequences for \(p_1,\cdots , p_s.\) Define
$$\begin{aligned} \begin{aligned} P_i(\alpha )&:= k_{i,1}\cdots k_{i,\alpha _i-1} \prod _{1\le i_0\le s,\> i_0\ne i} p_{i_0,1}\cdots p_{i_0,\alpha _{i_0-1}} \bmod {p_{i,\alpha _i}} \\ T(N)&:= \left\{ \alpha = (\alpha _1,\cdots ,\alpha _s)\in {\mathbb {N}}_0^s :\prod _{i=1}^s p_{i,1}\cdots p_{i,\alpha _i}\le N \right\} \\ M(\alpha )&:= \left\{ \ell = (\ell _1,\cdots ,\ell _s)\in {\mathbb {Z}}_{p_{1,\alpha _1}} \times \cdots \times {\mathbb {Z}}_{p_{s,\alpha _s}} :\ell _1+\cdots + \ell _s >0 \right\} \\ R(\alpha ;\ell )&:= \prod _{i=1}^s \max (1,\min (2\ell _i, 2(p_{i,\alpha _i} - \ell _i))) \end{aligned} \end{aligned}$$
for all \(1\le i\le s,\) \(\alpha = (\alpha _1,\cdots ,\alpha _s)\in {\mathbb {N}}_0^s,\) \(N\in {\mathbb {N}}\) and \(\ell = (\ell _1,\cdots , \ell _s)\in {\mathbb {N}}_0^s,\) where we denote \(k_{i,-1} = k_{i,0} = p_{i,-1}=p_{i,0} := 1.\)

In the following proposition, we formulate an estimate of discrepancy of the modified Halton sequence in Cantor bases which is the basis for our proof of Theorem 2. It can also be used for computational estimation of \(D^*_N(\omega ),\) if performing \(O((\log N)^s)\) operations is not a problem.

Proposition 1

Let \(\omega \) be the modified Halton sequence in Cantor bases \(p_1,\cdots , p_s\) with respect to \(k_1,\cdots , k_s.\) Then, for any \(N\in {\mathbb {N}},\) we have
$$\begin{aligned} ND^*_N(\omega )\le & {} \sum _{\alpha \in T(N)} \left( 1 + \sum _{\ell \in M(\alpha )} \frac{\left\| \sum _{i=1}^s\frac{\ell _i}{p_{i,\alpha _i}} P_i(\alpha )\right\| ^{-1}}{2 R(\alpha ;\ell )}\right) \\&+ \sum _{l =0}^{s-1} \frac{M_{l+1}}{l!} \prod _{i=1}^l\left( \frac{\lfloor M_i/2\rfloor \log N}{\log m_i} + l\right) , \end{aligned}$$
where \(M_i = \max (p_{i,j})_{j=1}^\infty \) and \(m_i = \min (p_{i,j})_{j=1}^\infty \) \((1\le i\le s),\) and where \(\Vert \cdot \Vert \) is the to-the-nearest-integer function.

The proof of this proposition is based on specific periodicity properties of the modified Halton sequence in Cantor bases. These properties will be studied in Lemma 8. Note that the following lemma appears in [1] in slightly different notation, and it is used to derive some properties in Lemma 8.

Lemma 7

[1, Lemma 4.2] Let \(p_{1,\alpha _1},\cdots , p_{s,\alpha _s}\) be distinct prime numbers, and let \(\xi = (\xi _{1,t}, \cdots , \xi _{s,t})_{t=0}^\infty \) be a sequence in \({\mathbb {Z}}^s.\) Let v and w be fixed integer s-tuples such that \(0\le v_i <w_i\le p_{i,\alpha _i}\) \((1\le i\le s).\) For each \(K\in {\mathbb {N}},\) we denote by
$$\begin{aligned} A_K(v,w) = \# \{0\le n\le K-1 :\forall \, 1\le i\le s,\,\, v_i\le \xi _{i,t}\bmod {p_{i,\alpha _i}} \le w_i-1\} \end{aligned}$$
the number of the first K terms of \(\xi \) such that, for all \(1\le i\le s,\) the remainder of \(\xi _{i,n}\) modulo \(p_{i,\alpha _i}\) is among the numbers \(v_i,\cdots , w_i-1.\) Then, for all \(K\in {\mathbb {N}},\) we have
$$\begin{aligned} \sup _{v,w} \left| A_K(v,w) - K\prod _{i=1}^s\frac{w_i-v_i}{p_{i,\alpha _i}} \right| \le \sum _{\ell \in M(\alpha )} \frac{|S_K(\ell ;\xi )|}{R(\alpha ;\ell )}, \end{aligned}$$
where
$$\begin{aligned} S_K(\ell ;\xi ) := \sum _{t=0}^{K-1} e\left( \sum _{i=1}^s \frac{\ell _i\xi _{i,t}}{p_{i,\alpha _i}}\right) , \end{aligned}$$
with the usual notation \(e(x) := \exp (2\pi i x).\)

Lemma 8

Let \(\omega \) be the modified Halton sequence in Cantor bases \(p_1,\cdots , p_s\) with respect to \(k_1,\cdots , k_s.\) Let I be an elementary interval of the form
$$\begin{aligned} I = \prod _{i=1}^s \left[ \frac{u_i}{p_{i,1}\cdots p_{i,\alpha _i-1}}, \frac{u_i+1}{p_{i,1}\cdots p_{i,\alpha _i-1}} \right) \end{aligned}$$
with integers \(0\le u_i \le p_{i,1}\cdots p_{i,\alpha _i-1}-1\) and \(\alpha _i\in {\mathbb {N}}_0\) \((1\le i\le s),\) and let J be a subinterval of I of the form
$$\begin{aligned} J = \prod _{i=1}^s \left[ \frac{u_i}{p_{i,1}\cdots p_{i,\alpha _i-1}} + \frac{v_i}{p_{i,1}\cdots p_{i,\alpha _i}}, \frac{u_i}{p_{i,1}\cdots p_{i,\alpha _i-1}} + \frac{w_i}{p_{i,1}\cdots p_{i,\alpha _i}} \right) \end{aligned}$$
with integers \(0\le v_i <w_i \le p_{i,\alpha _i}\) \((1\le i\le s).\) There exists a nonnegative integer n with \(\omega _n\in I.\) Let \(n_0\) be the smallest integer such that \(\omega _{n_0}\in I.\) Suppose that \(\omega _{n_0}\) drops into the interval
$$\begin{aligned} \prod _{i=1}^s \left[ \frac{u_i}{p_{i,1}\cdots p_{i,\alpha _i-1}} + \frac{x_i}{p_{i,1}\cdots p_{i,\alpha _i}}, \frac{u_i}{p_{i,1}\cdots p_{i,\alpha _i-1}} + \frac{x_i+1}{p_{i,1}\cdots p_{i,\alpha _i}} \right) \end{aligned}$$
with \(0\le x_i\le p_{i,\alpha _i}-1\) \((1\le i\le s).\) Then the following statements are true.
  1. (1)

    \(n_0 < \prod _{i=1}^s p_{i,1}\cdots p_{i,\alpha _i-1},\) and the indices of the terms of \(\omega \) that drop into I are of the form \(n= n_0 + t\prod _{i=1}^s p_{i,1}\cdots p_{i,\alpha _i-1}\) for some \(t\in {\mathbb {N}}_0.\)

     
  2. (2)

    Suppose that \(n = n_0 + t\prod _{i=1}^s p_{i,1}\cdots p_{i,\alpha _i-1}\) with \(t\in {\mathbb {N}}_0.\) Then \(\omega _n\in J\) if and only if there exist \(l_1\in \{v_1,\cdots , w_1-1\},\cdots , l_s\in \{v_s,\cdots , w_s-1\}\) such that \(x_i + tP_i(\alpha ) \equiv l_i \pmod {p_{i,\alpha _i}}\) for all \(1\le i\le s.\)

     
  3. (3)
    Let \(\xi = (\xi _{1,t},\cdots ,\xi _{s,t})_{t=0}^\infty \) be the sequence in \({\mathbb {Z}}^s\) with \(\xi _{i,t} = x_i + t P_i(\alpha )\) for each \(1\le i\le s.\) Let \(N\in {\mathbb {N}},\) and let K be the largest integer such that \(n_0+(K-1)\prod _{i=1}^s p_{i,1}\cdots p_{i,\alpha _i-1} <N.\) Then we have
    $$\begin{aligned} | A(J;N;\omega ) - N\lambda _s(J)| < 1 + \sum _{\ell \in M(\alpha )} \frac{S_K(\ell ;\xi )}{R(\alpha ;\ell )}. \end{aligned}$$
     

Proof

For each \(n\in {\mathbb {N}}_0,\) we denote the \(p_i\)-adic expansion of n by
$$\begin{aligned} n = \sum _{j=1}^\infty n_{i,j}p_{i,1}\cdots p_{i,j-1}, \end{aligned}$$
where \(n_{i,j}\in {\mathbb {Z}}_{p_{i,j}}\) \((j\in {\mathbb {N}}).\) On the other hand, we write the expansion of \(u_i\) as
$$\begin{aligned} u_i = u_{i,\alpha _i-1} + u_{i,\alpha _i-2}p_{i,\alpha _i-1} + u_{i,\alpha _i-3}p_{i,\alpha _i-1}p_{i,\alpha _i-2} + \cdots + u_{i,1}p_{i,\alpha _i-1}\cdots p_{i,2}, \end{aligned}$$
where \(u_{i,\alpha _i-j}\in {\mathbb {Z}}_{p_{i,\alpha _i-j}}\) \((1\le j\le \alpha _i-1).\) We have
$$\begin{aligned} \omega = \left( \sum _{j=1}^\infty \frac{n_{1,j}k_{1,1}\cdots k_{1,j-1}\bmod {p_{1,j}}}{p_{1,1}\cdots p_{1,j}},\cdots , \sum _{j=1}^\infty \frac{n_{s,j}k_{s,1}\cdots k_{s,j-1}\bmod {p_{s,j}}}{p_{s,1}\cdots p_{s,j}} \right) _{n=0}^\infty \end{aligned}$$
and
$$\begin{aligned} I = \prod _{i=1}^s \left[ \sum _{j=1}^{\alpha _i-1} \frac{u_{i,j}}{p_{i,1}\cdots p_{i,j}}, \sum _{j=1}^{\alpha _i-1} \frac{u_{i,j}}{p_{i,1}\cdots p_{i,j}} + \frac{1}{p_{i,1}\cdots p_{i,\alpha _i-1}} \right) . \end{aligned}$$
Then the nth element \(\omega _n\) of the modified Halton sequence in Cantor bases is contained in I if and only, for all \(1\le i\le s\) and all \(1\le j\le \alpha _i-1,\)
$$\begin{aligned} n_{i,j}k_{i,1}\cdots k_{i,j-1} \equiv u_{i,j} \pmod {p_{i,j}}. \end{aligned}$$
It follows, by Lemma 1, that there exists exactly one \(n_0\) such that \(\omega _{n_0}\in I\) and \(0\le n_0 <\prod _{i=1}^s p_{i,1}\cdots p_{i,\alpha _i-1}.\) For each \(1\le i\le s\) and all \(t\in {\mathbb {N}}_0,\) the first \(\alpha _i-1\) digits of \(n_0+ t\prod _{i_0=1}^s p_{i_0,1}\cdots p_{i_0,\alpha _{i_0}-1}\) in the \(p_i\)-adic number system are the same as that of \(n_0.\) Therefore, \(\omega _n\in I\) is equivalent to \(n = n_0+ t\prod _{i_0=1}^s p_{i_0,1}\cdots p_{i_0,\alpha _{i_0}-1}\) for some \(t\in {\mathbb {N}}_0.\) This proves the first assertion.
Next, suppose that \(n = n_0 + t\prod _{i_0=1}^s p_{i_0,1}\cdots p_{i_0,\alpha _{i_0}-1}\) with \(t\in {\mathbb {N}}_0.\) For all \(1\le i\le s,\) we now look at the \(\alpha _i\)th digit of the \(p_i\)-adic expansion of n. Since the \(\alpha _i\)th digit of \(t\prod _{i_0=1}^s p_{i_0,1}\cdots p_{i_0,\alpha _{i_0}-1}\) in the \(p_i\)-adic expansion is
$$\begin{aligned} \left( t\prod _{i_0=1}^s p_{i_0,1}\cdots p_{i_0,\alpha _{i_0}-1}\right) _{i,\alpha _i} = t\prod _{\begin{array}{c} 1\le i_0<1 \\ i_0\ne i \end{array}} p_{i_0,1}\cdots p_{i_0,\alpha _{i_0}-1} \bmod {p_{i,\alpha _i}}, \end{aligned}$$
we have
$$\begin{aligned} \begin{aligned} \sigma _{i,\alpha _i} \left( \left( t\prod _{i_0=1}^s p_{i_0,1}\cdots p_{i_0,\alpha _{i_0}-1}\right) _{i,\alpha _i}\right)&= tk_{i,1}\cdots k_{i,\alpha _i-1}\\&\prod _{\begin{array}{c} 1\le i_0<s \\ i_0\ne i \end{array}} p_{i_0,1}\cdots p_{i_0,\alpha _{i_0}-1} \bmod {p_{i,\alpha _i}} \\&= tP_i(\alpha ) \bmod {p_{i,\alpha _i}}. \end{aligned} \end{aligned}$$
Moreover, we have \(\sigma _{i,\alpha _i}((n_0)_{i,\alpha _i}) = x_i.\) Due to the fact that all the permutations \(\sigma _{i,j}\) are isomorphisms, we obtain
$$\begin{aligned} \sigma _{i,\alpha _i}((n)_{i,\alpha _i}) = x_i + tP_i(\alpha ) \bmod {p_{i,\alpha _i}}. \end{aligned}$$
It follows immediately that \(\omega _n\in J\) if and only if, for all \(1\le i\le s,\)
$$\begin{aligned} v_i \le x_i + tP_i(\alpha ) \bmod {p_{i,\alpha _i}} \le w_i-1. \end{aligned}$$
This proves the second assertion.
By the definition of K, we observe that \(A(J;N;\omega ) = A_K(v,w).\) Also, it is not hard to check that
$$\begin{aligned} -1 + K \prod _{i=1}^s \frac{w_i-v_i}{p_{i,\alpha _i}}< N\lambda _s(J) < 1 + K \prod _{i=1}^s \frac{w_i-v_i}{p_{i,\alpha _i}}. \end{aligned}$$
Now it follows that
$$\begin{aligned} |A(J;N;\omega ) - N\lambda _s(J)| < 1 + \left| A_K(v,w) - K \prod _{i=1}^s \frac{w_i-v_i}{p_{i,\alpha _i}} \right| . \end{aligned}$$
By using Lemma 7, we immediately obtain the last assertion. \(\square \)
Proof (Proof of Proposition 1)     Let \(J= \prod _{i=1}^s [0,z_i)\subseteq [0,1)^s.\) We expand each \(z_i\) in the same way as in the proof of Theorem 1, and obtain the equality (3) for \(A(J;N;\omega ) - N\lambda _s(J).\) The estimate in (4) for \(\varLambda _2\) depends only on Lemma 2, so we can use it here too. We now investigate
$$\begin{aligned} \varLambda _1 = \sum _{\alpha \in T(N)} \epsilon _{\alpha _1,\cdots ,\alpha _s} (A(J_{\alpha _1,\cdots ,\alpha _s};N;\omega ) - N\lambda _s(J_{\alpha _1,\cdots ,\alpha _s})). \end{aligned}$$
Let \(\alpha =(\alpha _1,\cdots ,\alpha _s)\in T(N).\) The interval \(J_{\alpha _1,\cdots ,\alpha _s}\) is contained inside some elementary interval
$$\begin{aligned} I = \prod _{i=1}^s \left[ \frac{u_i}{p_{i,1}\cdots p_{i,\alpha _i-1}}, \frac{u_i+1}{p_{i,1}\cdots p_{i,\alpha _i-1}} \right) \end{aligned}$$
with \(0\le u_i\le p_{i,1}\cdots p_{i,\alpha _i-1} -1\) \((1\le i\le s).\) Consider the sequence \(\xi ,\) defined as in Lemma 8, such that \(\xi _{i,t} = x_i + tP_i(\alpha )\) \((1\le i\le s),\) where the integers \(x_i\) are determined by the first term of the modified Halton sequence \(\omega \) that drops into the interval I fits into the smaller interval
$$\begin{aligned} \prod _{i=1}^s \left[ \frac{u_i}{p_{i,1}\cdots p_{i,\alpha _i-1}} + \frac{x_i}{p_{i,1}\cdots p_{i,\alpha _i}}, \frac{u_i}{p_{i,1}\cdots p_{i,\alpha _i-1}} + \frac{x_i+1}{p_{i,1}\cdots p_{i,\alpha _i}} \right) \end{aligned}$$
From the last property of Lemma 8, it follows that
$$\begin{aligned} |A(J_{\alpha _1,\cdots ,\alpha _s};N;\omega ) - N\lambda _s(J_{\alpha _1,\cdots ,\alpha _s})| < 1 + \sum _{\ell \in M(\alpha )}\frac{|S_K(\ell ;\xi )|}{R(\alpha ;\ell )}, \end{aligned}$$
where K is the number of terms of \(\omega \) among the first N that drop into the interval I.
Now we have
$$\begin{aligned} \varLambda _1 \le \sum _{\alpha \in T(N)} \left( 1 + \sum _{\ell \in M(\alpha )}\frac{|S_K(\ell ;\xi )|}{R(\alpha ;\ell )} \right) . \end{aligned}$$
In order to accomplish the proof, we claim that
$$\begin{aligned} |S_K(\ell ;\xi )| \le \frac{1}{2}\left\| \sum _{i=1}^s \frac{\ell _i}{p_{i,\alpha _i}} P_i(\alpha ) \right\| ^{-1}. \end{aligned}$$
(8)
Let \(\theta = \sum _{i=1}^s\frac{\ell _i}{p_{i,\alpha _i}} P_i(\alpha ).\) Since \(p_{1,\alpha _1},\cdots , p_{s,\alpha _s}\) are pairwise coprime and, for each \(1\le i\le s,\) \(p_{i,\alpha _i}\not \mid P_i(\alpha ),\) we must have \(\Vert \theta \Vert \ne 0.\) The inequality (8) follows immediately from
$$\begin{aligned} \left| \sum _{t=0}^{K-1} e(t\theta + \vartheta ) \right| = \frac{\sin (\pi \Vert K\theta \Vert )}{\sin (\pi \Vert \theta \Vert )}\le \frac{1}{2\Vert \theta \Vert } \quad (\vartheta \in \mathbb {R}). \end{aligned}$$
This completes the proof of Proposition 1. \(\square \)

The following two lemmas help extend Proposition 1 to Theorem 2. The first result shows that the modified Halton sequence in Cantor bases possesses some particular periodicity properties, while the other one which is borrowed directly from [1] provides some technical estimate to be used with the first lemma.

Lemma 9

Let \(p_1= (\overline{p_{1,1},\cdots , p_{1,j_1}}),\) \(\cdots ,\) \(p_s = (\overline{p_{s,1},\cdots , p_{s,j_s}})\) be periodic sequences of distinct prime numbers such that, for each \(1\le i\le s\), there is a common primitive root modulo \(p_{i,1},\cdots , p_{i,j_i}.\) Let \(k_1 = (\overline{k_{1,1},\cdots , k_{1,j_1}}),\) \(\cdots ,\) \(k_s = (\overline{k_{s,1},\cdots , k_{s,j_s}})\) be admissible sequences for \(p_1,\cdots , p_s.\) Denote
$$\begin{aligned} K := \left( j_1\cdots j_s\right) \prod _{\gamma _1 = 1}^{j_1} \cdots \prod _{\gamma _s =1}^{j_s}\left( \prod _{i=1}^s \left( p_{i,\gamma _i}-1\right) \right) . \end{aligned}$$
For each \(\beta =(\beta _1,\cdots , \beta _s)\in {\mathbb {N}}_0^s,\) denote
$$\begin{aligned} U(\beta ):= \left\{ \alpha =(\alpha _1,\cdots ,\alpha _s)\in {\mathbb {N}}_0^s :\forall \, 1\le i\le s,\>\> \beta _i K\le \alpha _i <(\beta _i+1)K\right\} . \end{aligned}$$
Then, for any \(\beta =(\beta _1,\cdots ,\beta _s)\in {\mathbb {N}}_0^s,\) any \(1\le \gamma _i\le j_i\) \((1\le i\le s)\) and any \((b_1,\cdots , b_s)\in {\mathbb {Z}}_{p_{1,\gamma _1}}^*\times \cdots \times {\mathbb {Z}}_{p_{s,\gamma _s}}^*,\) we have
$$\begin{aligned}&\#\{\alpha \in U(\beta ) :\forall \, 1\le i\le s,\,\, P_i(\alpha ) = b_i \,\,\,\text{ and }\,\,\, \alpha _i\equiv \gamma _i \>\>(\bmod {j_i}) \} \\&\quad = \frac{K^s}{j_1\cdots j_s \prod _{i=1}^s (p_{i,\gamma _i}-1)}. \end{aligned}$$

Proof

First, we observe that there are \(K^s/(j_1\cdots j_s)\) elements in \(U(\beta )\) such that \(\alpha _i \equiv \gamma _i \pmod {j_i}\) for each \(1\le i\le s,\) and that there are only \(\prod _{i=1}^s (p_{i,\gamma _i}-1)\) distinct elements \((b_1,\cdots , b_s)\) in \({\mathbb {Z}}_{p_{1,\gamma _1}}^*\times \cdots \times {\mathbb {Z}}_{p_{s,\gamma _s}}^*.\) By Pigeonhole Principle, there exists \((b_1',\cdots , b_s')\in {\mathbb {Z}}_{p_{1,\gamma _1}}^*\times \cdots \times {\mathbb {Z}}_{p_{s,\gamma _s}}^*\) such that
$$\begin{aligned}&\#\{\alpha \in U(\beta ) :\forall \, 1\le i\le s,\,\, P_i(\alpha ) = b_i' \,\,\,\text{ and }\,\,\, \alpha _i\equiv \gamma _i \>\>(\bmod {j_i}) \}\\&\quad \quad \ge \frac{K^s}{j_1\cdots j_s \prod _{i=1}^s (p_{i,\gamma _i}-1)}. \end{aligned}$$
For each \(1\le i\le s,\) let \(g_i\) be some fixed common primitive root modulo \(p_{i,1},\cdots , p_{i,j_i}.\) Since \(p_{i,\gamma _i}\not \mid b_i',\) the congruences \(b_i' \equiv g_i^{c_i} \pmod {p_{i,\gamma _i}}\) are fulfilled for some integers \(c_i\). Note from Lemma 6 that the equalities \(P_i(\alpha ) = b_i\) and the congruences \(\alpha _i\equiv \gamma _i \pmod {j_i}\) are possible if and only if \(\alpha \) together with some integers \(y_1,\cdots , y_s\) form a solution to the system (7). We conclude that if \(\alpha ',\alpha '' \in U(\beta )\) are two (possibly equal) solutions such that \(P_i(\alpha ') = b_i' = P_i(\alpha '')\) and \(\alpha _i' \equiv \gamma _i \equiv \alpha _i'' \pmod {j_i}\) for all \(1\le i\le s,\) then the s-tuple \(\alpha '''\) defined by
$$\begin{aligned} \alpha _i''' = \alpha _i' - \alpha _i'' + \gamma _i - \left[ \frac{\alpha _i' - \alpha _i'' + \gamma _i}{K}\right] \cdot K, \quad i=1,\cdots , s, \end{aligned}$$
is a (possibly trivial) solution of the congruences \(\alpha _i'''\equiv \gamma _i \pmod {j_i}\) and of the equations \(P_i(\alpha ''') = b_i^*,\) where \(b_i^* \equiv g_i^{\sum _{i_0}^s(a_{i,i_0,1}+\cdots + a_{i,i_0,\gamma _{i_0-1}})} \pmod {p_{i,\gamma _i}}\) with the notation from the system (7), and it is in U(0). It follows, from the choice of \((b_1',\cdots , b_s'),\) that
$$\begin{aligned}&\#\{\alpha \in U(0) :\forall \, 1\le i\le s,\,\, P_i(\alpha ) = b_i^* \,\,\,\text{ and }\,\,\, \alpha _i\equiv \gamma _i \>\>(\bmod {j_i}) \} \\&\quad \ge \frac{K^s}{j_1\cdots j_s \prod _{i=1}^s (p_{i,\gamma _i}-1)}. \end{aligned}$$
Let \((b_1,\cdots , b_s)\in {\mathbb {Z}}_{p_{1,\gamma _1}}^*\times \cdots \times {\mathbb {Z}}_{p_{s,\gamma _s}}^*.\) Since \(k_1,\cdots , k_s\) are admissible, we have
$$\begin{aligned} \#\{\alpha \in U(\beta ) :\forall \, 1\le i\le s,\,\, P_i(\alpha ) = b_i \,\,\,\text{ and }\,\,\, \alpha _i\equiv \gamma _i \>\>(\bmod {j_i}) \} \ge 1. \end{aligned}$$
Let \(\alpha \in U(\beta )\) be such that \(P_i(\alpha ) = b_i\) and \(\alpha _i\equiv \gamma _i\pmod {j_i}\) for each \(1\le i\le s,\) and let \(\alpha ^0\in U(0)\) be such that \(P_i(\alpha ^0) = b_i^*\) and \(\alpha _i^0\equiv \gamma _i\pmod {j_i}\) for each i. Then the s-tuple \(\alpha ^*\) defined by
$$\begin{aligned} \alpha _i^* = \alpha _i + \alpha _i^0 - \gamma _i -\left[ \frac{\alpha _i + \alpha _i^0 - \gamma _i}{K}\right] \cdot K + \beta _i K, \quad i = 1,\cdots , s, \end{aligned}$$
yields another solution of the congruences \(\alpha _i^*\equiv \gamma _i\pmod {j_i}\) and of the equations \(P_i(\alpha ^*) = b_i,\) and it is in U(x). It follows immediately that
$$\begin{aligned}&\#\{\alpha \in U(\beta ) :\forall \, 1\le i\le s,\,\, P_i(\alpha ) = b_i \,\,\,\text{ and }\,\,\, \alpha _i\equiv \gamma _i \>\>(\bmod {j_i}) \}\\&\quad \ge \frac{K^s}{j_1\cdots j_s \prod _{i=1}^s (p_{i,\gamma _i}-1)} \end{aligned}$$
since there are at least \(K^s/(j_1\cdots j_s \prod _{i=1}^s(p_{i,\gamma _i}-1))\) such \(\alpha _i^0.\) Because this is true for all \((b_1,\cdots , b_s),\) it follows that the number of the solutions in \(U(\beta )\) is exactly \(K^s/(j_1\cdots j_s \prod _{i=1}^s(p_{i,\gamma _i}-1)).\) This completes the proof of Lemma 9. \(\square \)

Lemma 10

[1, Lemma 4.4] Let \(p_1,\cdots , p_s\) be distinct prime numbers. Then
$$\begin{aligned}&\sum _{\ell \in M(p_1,\cdots , p_s)} \sum _{b_1=1}^{p_1-1} \cdots \sum _{b_s = 1}^{p_s-1} \frac{\left\| \frac{\ell _1 b_1}{p_1} + \cdots + \frac{\ell _s b_s}{p_s}\right\| ^{-1}}{2 R(p_1,\cdots , p_s;\ell )}\\&\quad \le \left( \sum _{i=1}^s\log p_i \right) \left( \prod _{i=1}^s p_i \right) \left( -1+\prod _{i=1}^s (1+\log p_i) \right) , \end{aligned}$$
where we denote
$$\begin{aligned} \begin{aligned} M(p_1,\cdots , p_s)&:= \left\{ \ell =(\ell _1,\cdots ,\ell _s)\in {\mathbb {Z}}_{p_1}\times {\mathbb {Z}}_{p_s} :\ell _1+\cdots + \ell _s >0 \right\} , \\ R(p_1,\cdots , p_s;\ell )&:= \prod _{i=1}^s \max (1,\min (2\ell _i,2(p_i-\ell _i))). \end{aligned} \end{aligned}$$

Now we are in a position to prove our second main theorem.

Proof (Proof of Theorem 2)    Our proof is based upon Proposition 1. Let
$$\begin{aligned} K = \left( j_1\cdots j_s\right) \prod _{\gamma _1 = 1}^{j_1} \cdots \prod _{\gamma _s =1}^{j_s}\left( \prod _{i=1}^s \left( p_{i,\gamma _i}-1\right) \right) . \end{aligned}$$
For each \(\beta =(\beta _1,\cdots , \beta _s)\in {\mathbb {N}}_0^s,\) denote
$$\begin{aligned} U(\beta ) := \left\{ \alpha =(\alpha _1,\cdots ,\alpha _s)\in {\mathbb {N}}_0^s :\forall \, 1\le i\le s,\>\> \beta _i K\le \alpha _i <(\beta _i+1)K\right\} . \end{aligned}$$
Clearly, each \(\alpha = (\alpha _1,\cdots ,\alpha _s)\in T(N)\) is inside a unique box \(U(\beta )\) such that the s-tuple \(\beta \) satisfies \(\prod _{i=1}^s p_{i,1}\cdots p_{i,\beta _iK} \le N.\) We apply Lemma 3, for the integers \(b_{i,j} := (p_{i,1}\cdots p_{i,j_i})^{K/j_i}\) and the bounds \(f_i:=1,\) to obtain an estimate of the number of those boxes \(U(\beta )\) which contain T(N),  i.e. we have
$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} \beta =(\beta _1,\cdots ,\beta _s) \in {\mathbb {N}}_0^s \\ \prod _{i=1}^s p_{i,1}\cdots p_{i,\beta _iK} \le N \end{array}} 1&= \sum _{\begin{array}{c} \beta =(\beta _1,\cdots ,\beta _s) \in {\mathbb {N}}_0^s \\ \prod _{i=1}^s \left( \left( p_{i,1}\cdots p_{i,j_i}\right) ^{K/j_i}\right) ^{\beta _i} \le N \end{array}} 1 \\&\le \frac{1}{s!}\prod _{i=1}^s\left( \frac{\log N}{(K/j_i)\log (p_{i,1}\cdots p_{i,j_i}) } + s \right) \\&\le \frac{1}{s!}\prod _{i=1}^s\left( \frac{\log N}{K \log m_i} + s \right) . \end{aligned} \end{aligned}$$
Now we can use Lemma 9 to obtain a partial estimate of the first sum in the inequality in Proposition 1 as follows
$$\begin{aligned} \begin{aligned}&\sum _{\alpha \in T(N)} \sum _{\ell \in M(\alpha )}\frac{\left\| \sum _{i=1}^s \frac{\ell _i}{p_{i,\alpha _i}} P_i(\alpha ) \right\| ^{-1}}{2R(\alpha ;\ell )} \\&\quad \le \sum _{\begin{array}{c} \beta \in {\mathbb {N}}_0^s \\ \prod _{i=1}^s p_{i,1}\cdots p_{i,\beta _iK} \le N \end{array}} \left( \sum _{\gamma _1=1}^{j_1} \cdots \sum _{\gamma _s=1}^{j_s} \frac{K^s}{j_1\cdots j_s \prod _{i=1}^s(p_{i,\gamma _i}-1)} \left( \sum _{\ell \in M(p_{1,\gamma _1},\cdots , p_{s,\gamma _s})} \right. \right. \\&\qquad \left. \left. \sum _{b_1=1}^{p_{1,\gamma _1}-1}\cdots \sum _{b_s=1}^{p_{s,\gamma _s}-1} \frac{\left\| \sum _{i=1}^s \frac{\ell _i b_i}{p_{i,\gamma _i}} \right\| ^{-1}}{2R(p_{1,\gamma _1},\cdots , p_{s,\gamma _s};\ell )} \right) \right) . \end{aligned} \end{aligned}$$
We apply Lemma 10 to the rightmost sums of the above inequality to get
$$\begin{aligned} \begin{aligned}&\sum _{\alpha \in T(N)} \sum _{\ell \in M(\alpha )}\frac{\left\| \sum _{i=1}^s \frac{\ell _i}{p_{i,\alpha _i}} P_i(\alpha ) \right\| ^{-1}}{2R(\alpha ;\ell )} \le \sum _{\begin{array}{c} \beta \in {\mathbb {N}}_0^s \\ \prod _{i=1}^s p_{i,1}\cdots p_{i,\beta _iK} \le N \end{array}}\\&\left( \sum _{\gamma _1=1}^{j_1} \cdots \sum _{\gamma _s=1}^{j_s} \frac{K^s}{j_1\cdots j_s \prod _{i=1}^s(p_{i,\gamma _i}-1)} \left( \left( \sum _{i=1}^s \log p_{i,\gamma _i}\right) \left( \prod _{i=1}^s p_{i,\gamma _i}\right) \right. \right. \\&\qquad \left. \left. \left( -1 + \prod _{i=1}^s (1+\log p_{i,\gamma _i})\right) \right) \right) \le \sum _{\begin{array}{c} \beta \in {\mathbb {N}}_0^s \\ \prod _{i=1}^s p_{i,1}\cdots p_{i,\beta _iK} \le N \end{array}} \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned}&\quad \frac{K^s}{\prod _{i=1}^s(m_i-1)} \left( \sum _{i=1}^s \log M_i\right) \left( \prod _{i=1}^s M_i\right) \left( -1 + \prod _{i=1}^s (1+\log M_i)\right) \\&\quad \le \frac{1}{s!}\left( \prod _{i=1}^s\left( \frac{\log N}{K \log m_i} + s \right) \right) \left( \frac{K^s}{\prod _{i=1}^s(m_i-1)} \right) \left( \sum _{i=1}^s \log M_i\right) \\&\qquad \left( \prod _{i=1}^s M_i\right) \left( -1 + \prod _{i=1}^s (1+\log M_i)\right) \end{aligned} \end{aligned}$$
(9)
To obtain the estimate of the other part of the first sum in the inequality in Proposition 1, we utilize Lemma 3 again, that is, we have
$$\begin{aligned} \sum _{\alpha \in T(N)} 1 = \sum _{\begin{array}{c} \alpha \in {\mathbb {N}}_0^s \\ \prod _{i=1}^s p_{i,1}\cdots p_{i,\alpha _i} \le N \end{array}} 1 \le \frac{1}{s!} \prod _{i=1}^s\left( \frac{\log N}{\log m_i} +s\right) . \end{aligned}$$
(10)
We finally combine Proposition 1 with () and (10) to obtain
$$\begin{aligned} \begin{aligned} ND^*_N(\omega )&\le \left( \frac{1}{s!} \left( \frac{1}{\prod _{i=1}^s\log m_i}\right) \right. \\&\quad \left. \left( 1+ \frac{\sum _{i=1}^s\log M_i \prod _{i=1}^s M_i \left( -1 + \prod _{i=1}^s(1+\log M_i)\right) }{\prod _{i=1}^s (m_i-1)}\right) \right) (\log N)^s \\&\quad + O\left( (\log N)^{s-1}\right) \\&\le \left( \frac{1}{s!} \left( \sum _{i=1}^s \log M_i\right) \prod _{i=1}^s \frac{M_i(1+\log M_i)}{(m_i-1)\log m_i} \right) (\log N)^s + O\left( (\log N)^{s-1}\right) , \end{aligned} \end{aligned}$$
where the last inequality follows from \((\sum _{i=1}^s \log M_i)(\prod _{i=1}^s M_i/(m_i-1)) \ge 1.\) This completes the proof of Theorem 2. \(\square \)

6 The generalized Hammersley point set in Cantor bases

Based on the \((s-1)\)-dimensional generalized Halton sequence, we can introduce a finite s-dimensional point set which is called the generalized Hammersley point set.

Let \(b_1=(b_{1,j})_{j=1}^\infty ,\) \(\cdots ,\) \(b_{s-1}=(b_{s-1,j})_{j=1}^\infty \) be \(s-1\) sequences of natural numbers greater than 1 such that, for all \(1\le i_1 < i_2 \le s-1\) and all \(j_1,j_2\in {\mathbb {N}},\) \(b_{i_1,j_1}\) and \(b_{i_2,j_2}\) are coprime. For each \(1\le i\le s-1,\) let \(\varSigma _i = (\sigma _{i,j})_{j=1}^\infty \) be an arbitrary sequence of permutations of \({\mathbb {Z}}_{b_{i,j}}\) \((j\in {\mathbb {N}}).\) The generalized Hammersley point set in Cantor bases \(b_1,\cdots , b_{s-1}\) with respect to \(\varSigma _1,\cdots ,\varSigma _{s-1},\) consisting of N points in \([0,1)^s,\) is defined to be the point set
$$\begin{aligned} {\mathcal {P}} = \left\{ \left( \frac{n}{N}, \phi _{b_1}^{\varSigma _1}(n),\cdots , \phi _{b_{s-1}}^{\varSigma _{s-1}}(n)\right) :0\le n\le N-1 \right\} . \end{aligned}$$
We deduce a discrepancy bound for the generalized Hammersley point set in Cantor bases with the help of Theorem 1 in combination with the following general result from [19] that goes back to Roth [20].

Lemma 11

[19, Lemma 3.7] Let \(\omega = (x_n)_{n=0}^\infty \) be an arbitrary sequence in \([0,1)^{s-1}\) with discrepancy \(D_N^*(\omega ).\) For \(N\in {\mathbb {N}},\) let \({\mathcal {P}}\) be the point set consisting of \((n/N, x_n)\) in \([0,1)^s\) for \(n=0,1,\cdots , N-1.\) Then we have
$$\begin{aligned} N D_N^*({\mathcal {P}}) \le \max _{1\le N'\le N} N' D_{N'}^*(\omega ) +1. \end{aligned}$$

Theorem 3

Let \(b_1 = (b_{1,j})_{j=1}^\infty , \cdots , b_{s-1} = (b_{s-1,j})_{j=1}^\infty \) be \(s-1\) arbitrary sequences of natural numbers greater than 1 such that, for all \(1\le i_1 < i_2 \le s-1\) and all \(j_1,j_2\in {\mathbb {N}},\) \(b_{i_1,j_1}\) and \(b_{i_2,j_2}\) are coprime. For each \(1\le i\le s-1\) and \(j\in {\mathbb {N}},\) let \(\sigma _{i,j}\) be a permutation of \({\mathbb {Z}}_{b_{i,j}}.\) For each \(1\le i\le s,\) denote \(\varSigma _i = (\sigma _{i,j})_{j=1}^\infty .\) For each \(N\in {\mathbb {N}},\) suppose that \({\mathcal {P}}\) is the generalized Hammersley point set in Cantor bases \(b_1,\cdots , b_{s-1}\) with respect to \(\varSigma _1,\cdots ,\varSigma _{s-1}\) consisting of N points. Then, for any \(N\ge 1,\) we have
$$\begin{aligned} N D^*_N({\mathcal {P}})\le & {} \frac{1}{(s-1)!} \prod _{i=1}^{s-1} \left( \frac{\lfloor M_i/2\rfloor \log N}{\log m_i} + s-1 \right) \\&+ \sum _{l =0}^{s-2} \frac{M_{l+1}}{l!} \prod _{i=1}^l\left( \frac{\lfloor M_i/2\rfloor \log N}{\log m_i} + l\right) + 1, \end{aligned}$$
where \(M_i = \max \{b_{i,j}\in b_i :b_{i,1}\cdots b_{i,j} \le N\}\) and \(m_i = \min \{b_{i,j}\in b_i :b_{i,1}\cdots b_{i,j} \le N\}\) \((1\le i\le s-1).\)

Corollary 1

Let \(b_1 = (b_{1,j})_{j=1}^\infty , \cdots , b_{s-1} = (b_{s-1,j})_{j=1}^\infty \) be \(s-1\) bounded sequences of natural numbers greater than 1 such that, for all \(1\le i_1 < i_2 \le s-1\) and all \(j_1,j_2\in {\mathbb {N}},\) \(b_{i_1,j_1}\) and \(b_{i_2,j_2}\) are coprime. For each \(1\le i\le s-1\) and \(j\in {\mathbb {N}},\) let \(\sigma _{i,j}\) be a permutation of \({\mathbb {Z}}_{b_{i,j}}.\) For each \(1\le i\le s,\) denote \(\varSigma _i = (\sigma _{i,j})_{j=1}^\infty .\) For each \(N\in {\mathbb {N}},\) suppose that \({\mathcal {P}}\) is the generalized Hammersley point set in Cantor bases \(b_1,\cdots , b_{s-1}\) with respect to \(\varSigma _1,\cdots ,\varSigma _{s-1}\) consisting of N points. Then, for any \(N\ge 1,\) we have
$$\begin{aligned} N D^*_N({\mathcal {P}}) \le c \frac{(\log N)^{s-1}}{N} + O\left( \frac{(\log N)^{s-2}}{N}\right) \end{aligned}$$
with
$$\begin{aligned} c=c(b_1,\cdots , b_{s-1}) = \frac{1}{(s-1)!} \prod _{i=1}^{s-1} \frac{\lfloor M_i/2\rfloor }{\log m_i}, \end{aligned}$$
where \(M_i = \max (b_{i,j})_{j=1}^\infty \) and \(m_i = \min (b_{i,j})_{j=1}^\infty \) \((1\le i\le s-1).\)

A point set \({\mathcal {P}}\) consisting of N points in \([0,1)^s\) is called a low-discrepancy point set if \(D_N^*({\mathcal {P}}) = O((\log N)^{s-1}/N).\) In this sense, the generalized Hammersley point set in Cantor bases is a low-discrepancy point set.

Notes

Acknowledgements

The authors wish to express their gratitude to the anonymous referee for thorough reading of the manuscript and for many valuable remarks which substantially improve the paper. In addition, the second author would like to thank Prof. Alev Topuzoǧlu at Sabancı Üniversitesi for a useful discussion.

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of Hradec KrálovéHradec KrálovéCzech Republic
  2. 2.Instytut MatematykiUniwersytet GdańskiGdańskPoland
  3. 3.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil

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