# The generalized and modified Halton sequences in Cantor bases

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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

*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

*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.

*n*has a unique

*b*-adic representation of the form

*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

*N*elements of the Halton sequence in bases \(b_1,\cdots , b_s\) can be bounded by

*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

*s*prime numbers, \(c(b_1,\cdots , b_s)\rightarrow 0\) as \(s\rightarrow \infty .\)

*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

*n*has a unique

*b*-adic representation of the form

*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

*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].

*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

*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

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].

## 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

*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

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

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

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

*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

### Proof

*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

*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 \(n=n'.\)

*n*. Define \(F:{\mathbb {Z}}_{b^*_1\cdots b^*_s} \rightarrow {\mathbb {Z}}_{b^*_1}\times \cdots \times {\mathbb {Z}}_{b^*_s}\) by

*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

*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

### Proof

*n*by

*n*th element \(\omega _n\) of the generalized Halton sequence in Cantor bases is contained in \(J_\ell \) if and only if, for all \(1\le i\le s,\)

*J*as a disjoint union of intervals of the form \(J_\ell ,\)

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

### Lemma 3

*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

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 \(\frac{1}{s!}\prod _{i=1}^s \frac{\log N}{\log m_i}.\)

*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

*A*,

*B*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

*J*. The following lemma is borrowed from [1] (see also [5] for a detailed proof).

### Lemma 4

*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

*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

*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

*J*.

*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 \(1\le l\le s-1,\)

*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

*L*is contained in the interval

## 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

*s*integer variables \(y_1,\cdots , y_s\) to change the congruences (6) into a system of Diophantine equations

*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:

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

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

*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

*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

### Lemma 8

*I*be an elementary interval of the form

*J*be a subinterval of

*I*of the form

*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

- (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)
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)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

*n*by

*n*th 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 \(1\le j\le \alpha _i-1,\)

*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

*K*, we observe that \(A(J;N;\omega ) = A_K(v,w).\) Also, it is not hard to check that

*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

*I*fits into the smaller interval

*K*is the number of terms of \(\omega \) among the first

*N*that drop into the interval

*I*.

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

### Proof

*s*-tuple \(\alpha '''\) defined by

*U*(0). It follows, from the choice of \((b_1',\cdots , b_s'),\) that

*i*. Then the

*s*-tuple \(\alpha ^*\) defined by

*U*(

*x*). It follows immediately that

### Lemma 10

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

*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

## 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.

*N*points in \([0,1)^s,\) is defined to be the point set

### Lemma 11

### Theorem 3

*N*points. Then, for any \(N\ge 1,\) we have

### Corollary 1

*N*points. Then, for any \(N\ge 1,\) we have

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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