On a bounded remainder set for $(t,s)$ sequences I

TL;DR

This paper characterizes bounded remainder sets for certain low-discrepancy sequences in multi-dimensional unit cubes, showing that these sets are precisely those with finitely many non-zero digits in the base expansion of their defining parameters.

Contribution

It provides a complete characterization of bounded remainder sets for Halton-type and related sequences derived from global function fields, extending previous results to a broader class of sequences.

Findings

Bounded remainder sets are characterized by finitely many non-zero digits in the base expansion.

The results apply to Halton-type, Niederreiter, Xing-Niederreiter, and Niederreiter-Xing sequences.

The characterization is both necessary and sufficient for these sequences.

Abstract

Let ${\bf x}_0,{\bf x}_1,...$ be a sequence of points in $[0,1)^s$. A subset $S$ of $[0,1)^s$ is called a bounded remainder set if there exist two real numbers $a$ and $C$ such that, for every integer $N$, $$ | {\rm card}\{n <N \; | \; {\bf x}_{n} \in S\} - a N| <C . $$ Let $ ({\bf x}_n)_{n \geq 0} $ be an $s-$dimensional Halton-type sequence obtained from a global function field, $b \geq 2$, ${\bf \gamma} =(\gamma_1,...,\gamma_s)$, $\gamma_i \in [0, 1)$, with $b$-adic expansion $\gamma_i= \gamma_{i,1}b^{-1}+ \gamma_{i,2}b^{-2}+...$, $i=1,...,s$. In this paper, we prove that $[0,\gamma_1) \times ...\times [0,\gamma_s)$ is the bounded remainder set with respect to the sequence $({\bf x}_n)_{n \geq 0}$ if and only if \begin{equation} \nonumber \max_{1 \leq i \leq s} \max \{ j \geq 1 \; | \; \gamma_{i,j} \neq 0 \} < \infty. \end{equation} We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.

Equations

$$|\mathrm{card}\{n<N|{\mathbf{x}}_n\in S\}-aN|<C.$$

$$\underset{1\le i\le s}{\max }\max \{j\ge 1|{\gamma }_{i,j}\ne 0\}<\infty .$$

$$\Delta (S,({\mathbf{x}}_n{)}_{n=0}^{N-1})=\sum _{n=0}^{N-1}(1_S({\mathbf{x}}_n)-\lambda (S)),$$

$${\text{D}}^{\ast }(({\mathbf{x}}_n{)}_{n=0}^{N-1})={\sup }_{0<y_1,\dots ,y_s\le 1}|\Delta ([0,\mathbf{y}),({\mathbf{x}}_n{)}_{n=0}^{N-1})/N|,$$

$$\underset{N\to \infty }{\mathrm{limsup}}N(\ln N{)}^{-s}{\text{D}}^{\ast }(({\mathbf{x}}_n{)}_{n=0}^{N-1})>0.$$

$$\underset{1\le i\le s}{\max }\max \{j\ge 1|{\gamma }_{i,j}\ne 0\}<\infty .$$

$$E=\prod _{i=1}^s[a_i{b}^{-d_i},(a_i+1)b^{-d_i}),$$

$$\mathbf{n}=({\bar{a}}_0(n),...,{\bar{a}}_{m-1}(n))\in {\mathbb{F}}_b^m\mathrm{with}{\bar{a}}_r(n)=\varphi (a_r(n)),r\in [0,m).$$

$$y_n^{(i)}=(y_{n,1}^{(i)},...,y_{n,m}^{(i)}):=\mathbf{n}{C}^{(i,m)\top },y_{n,j}^{(i)}=\sum _{r=0}^{\infty }{\bar{a}}_r(n)c_{j,r}^{(i)}\in {\mathbb{F}}_b,$$

$$x_n^{(i)}=\sum _{j=1}^m{x}_{n,j}^{(i)}/b^j,x_{n,j}^{(i)}={\varphi }^{-1}(y_{n,j}^{(i)})$$

$${\mathbf{x}}_n:=(x_n^{(1)},...,x_n^{(s)})\in [0,1{)}^s.$$

$${\mathcal{N}}^{\perp }=\{A\in {\mathbb{F}}_b^{sm}|B\cdot A=0\mathrm{for}\mathrm{all}B\in \mathcal{N}\}.$$

$$[C{]}_m=([C^{(1)}{]}_m^{\top }|[C^{(2)}{]}_m^{\top }|...|[C^{(s)}{]}_m^{\top })\in {\mathbb{F}}_b^{m\times sm}$$

$$\mathcal{C}_{m}=\Big{\{}\Big{(}\sum_{r=0}^{m-1}c^{(i)}_{j,r}\bar{a}_{r}(n)\Big{)}_{1\leq j\leq m,1\leq i\leq s}\;|\;0\leq n<b^{m}\Big{\}}.$$

$${\mathcal{C}}_m^{\perp }=\{A\in {\mathbb{F}}_b^{sm}|B\cdot A^{\top }=0\mathrm{for}\mathrm{all}B\in {\mathcal{C}}_m\}.$$

$$\underset{0\le k<n<b^m}{\min }{\Vert {\mathbf{x}}_n\ominus {\mathbf{x}}_k\Vert }_b>b^{-m-d}\mathrm{where}{\Vert \mathbf{x}\Vert }_b:=\prod _{i=1}^s{\Vert x_j^{(i)}\Vert }_b.$$

$$\underset{1\le N\le b^m}{\max }N{\text{D}}^{\ast }(({\mathbf{x}}_n\oplus \mathbf{w}{)}_{0\le n<N})\ge K{m}^s$$

$${\varkappa }_m:=\underset{0\le k<n<b^m}{\min }{\Vert {\mathbf{x}}_n\ominus {\mathbf{x}}_k\Vert }_b>0\forall m\ge 1\mathrm{where}{\Vert \mathbf{x}\Vert }_b:=\prod _{i=1}^s{\Vert x^{(i)}\Vert }_b.$$

$$g_{\mathbf{w}}=\{A\ge 1|x_{b^{m}A,j}^{(i)}=w_j^{(i)},j\in [1,{\tau }_m],i\in [1,s]\}\mathrm{and}{g}_{\mathbf{w}}\ne \varnothing \forall w_j^{(i)}\in {\mathbb{Z}}_b.$$

$${\nu }_{\infty }(\alpha )=\mathrm{deg}(g)-\mathrm{deg}(f)$$

$$\mathbb{F}_{b}((x)):=\Big{\{}\sum_{i=m}^{\infty}a_{i}x^{i}\;|\;m\in\mathbb{Z},\;a_{i}\in\mathbb{F}_{b}\Big{\}}.$$

$$\alpha =\sum _{i={\nu }_{\mathcal{P}}(\alpha )}^{\infty }{S}_i{t}^i\mathrm{where}{S}_i=S_i(t,\alpha )\in F_{\mathcal{P}}\subseteq F^{(P)}.$$

$$\frac{\mathrm{d}\alpha }{\mathrm{d}t}=\sum _{i={\nu }_{\mathcal{P}}(\alpha )}^{\infty }i{S}_i{t}^{i-1}.$$

$${\Delta }_F=\{y\mathrm{d}z|y\in F,z\mathrm{is}\mathrm{a}\mathrm{separating}\mathrm{element}\mathrm{for}F/{\mathbb{F}}_b\}$$

$${\nu }_{\mathcal{P}}(\alpha \mathrm{d}\beta ):={\nu }_{\mathcal{P}}(\alpha \mathrm{d}\beta /\mathrm{d}t),$$

$$\mathrm{deg}(\dot{W})=2g-2\mathrm{and}\ell (\dot{W})=g.$$

$${\nu }_{\mathcal{P}}(\mathrm{div}(\omega ))={\nu }_{\mathcal{P}}(\alpha \mathrm{d}\beta ),\mathrm{for}\mathrm{all}\mathcal{P}\in {\mathbb{P}}_F.$$

$$\alpha \mathrm{d}\beta =\sum _{i={\nu }_{\mathcal{P}}(\alpha )}^{\infty }{S}_i{t}^i\mathrm{d}t\in F^{(\mathcal{P})}.$$

$${\mathrm{Res}}_{\mathcal{P}}(\alpha \mathrm{d}\beta ):={\mathrm{Tr}}_{F_{\mathcal{P}}/{\mathbb{F}}_b}(S_{-1})\in {\mathbb{F}}_b.$$

$${\mathrm{Res}}_{\mathcal{P},t}(\alpha ):={\mathrm{Res}}_{\mathcal{P}}(\alpha \mathrm{d}t).$$

Full text

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Let $({\bf x}_{n})_{n\geq 0}$ be an $s-$dimensional Halton-type sequence obtained from a global function field, $b\geq 2$, $\boldsymbol{\gamma}=(\gamma_{1},...,\gamma_{s})$, $\gamma_{i}\in[0,1)$, with $b$-adic expansion $\gamma_{i}=\gamma_{i,1}b^{-1}+\gamma_{i,2}b^{-2}+...$, $i=1,...,s$. In this paper, we prove that $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is the bounded remainder set with respect to the sequence $({\bf x}_{n})_{n\geq 0}$ if and only if

[TABLE]

We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.

Dedicated to the 100th anniversary of Professor N.M. Korobov

Key words: bounded remainder set, $(t,s)$ sequence, Halton type sequences.

2010 Mathematics Subject Classification. Primary 11K38.

1 Introduction

1.1. Bounded remainder sets. Let ${\bf x}_{0},{\bf x}_{1},...$ be a sequence of points in $[0,1)^{s}$, $S\subseteq[0,1)^{s}$,

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where $\mathds{1}_{S}({\bf x})=1,\;{\rm if}\;{\bf x}\in S$, and $\mathds{1}_{S}({\bf x})=0,$ if ${\bf x}\notin S$. Here $\lambda(S)$ denotes the $s$-dimensional Lebesgue-measure of $S$. We define the star discrepancy of an $N$-point set $({\bf x}_{n})_{n=0}^{N-1}$ as

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where $[{\bf 0},{\bf y})=[0,y_{1})\times\cdots\times[0,y_{s})$. The sequence $({\bf x}_{n})_{n\geq 0}$ is said to be uniformly distributed in $[0,1)^{s}$ if $D_{N}\to\;0$. In 1954, Roth proved that $\limsup\nolimits_{N\to\infty}N(\ln N)^{-\frac{s}{2}}\emph{D}^{*}(({\bf x}_{n})_{n=0}^{N-1})>0.$ According to the well-known conjecture (see, e.g., [BeCh, p.283]), this estimate can be improved to

[TABLE]

See [Bi] and [Le1] for results on this conjecture.

A sequence $({\bf x}_{n}^{(s)})_{n\geq 0}$ is of low discrepancy (abbreviated l.d.s.) if $\emph{D}(({\bf x}_{n}^{(s)})_{n=0}^{N-1})\\ =O(N^{-1}(\ln N)^{s})$ for $N\rightarrow\infty$. A sequence of point sets $(({\bf x}_{n,N}^{(s)})_{n=0}^{N-1})_{N=1}^{\infty}$ is of low discrepancy (abbreviated l.d.p.s.) if $\emph{D}(({\bf x}_{n,N}^{(s)})_{n=0}^{N-1})=O(N^{-1}(\ln N)^{s-1})$, for $N\rightarrow\infty$. For examples of such a sequences, see, e.g., [BeCh], [DiPi], and [Ni].

Definition 1. *Let ${\bf x}_{0},{\bf x}_{1},...$ be a sequence of points in $[0,1)^{s}$. A subset $S$ of $[0,1)^{s}$ is called a bounded remainder set for $({\bf x}_{n})_{n\geq 0}$ if the discrepancy function $\Delta(S,({\bf x}_{n})_{n=0}^{N-1})$ is bounded in $\mathbb{N}$.

Let $\alpha$ be an irrational number, let I be an interval in $[0,1)$ with length $|I|$, let $\{n\alpha\}$ be the fractional part of $n\alpha$, $n=1,2,...$ . Hecke, Ostrowski and Kesten proved that $\Delta(S,(\{n\alpha\})_{n=1}^{N})$ is bounded if and only if $|I|=\{k\alpha\}$ for some integer $k$ (see references in [GrLe]).

The sets of bounded remainder for the classical $s$-dimensional Kronecker sequence studied by Lev and Grepstad [GrLe]. The case of Halton’s sequence was studied by Hellekalek [He].

Let $b$ be a prime power, $\boldsymbol{\gamma}=(\gamma_{1},...,\gamma_{s})$, $\gamma_{i}\in(0,1)$ with $b$-adic expansion $\gamma_{i}=\gamma_{i,1}b^{-1}+\gamma_{i,2}b^{-2}+...$, $i=1,...,s$, and let $({\bf x}_{n})_{n\geq 0}$ be a uniformly distributed digital Kronecker sequence. In [Le1], we proved the following theorem:

Theorem A. The set $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is of bounded remainder with respect to $({\bf x}_{n})_{n\geq 0}$ if and only if

[TABLE]

In this paper, we prove similar results for digital sequences described in [DiPi, Sec. 8]. Note that according to Larcher’s conjecture [La2, p.215], the assertion of Theorem A is true for all digital $(t,s)$-sequences in base $b$.

2 Definitions and auxiliary results.

2.1 $({\bf T},s)$ sequences. A subinterval $E$ of $[0,1)^{s}$ of the form

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with $a_{i},d_{i}\in\mathbb{Z},\;d_{i}\geq 0,\;0\leq a_{i}<b^{d_{i}}$, for $1\leq i\leq s$ is called an elementary interval in base $b\geq 2$.

Definition 2. Let $0\leq t\leq m$ be integers. A $(t,m,s)$-net in base $b$* is a point set ${\bf x}_{0},...,{\bf x}_{b^{m}-1}$ in $[0,1)^{s}$ such that $\#\{n\in[0,b^{m}-1]|x_{n}\in E\}=b^{t}$ for every elementary interval E in base $b$ with ${\rm vol}(E)=b^{t-m}$.

Definition 3. Let $t\geq 0$ be an integer. A sequence ${\bf x}_{0},{\bf x}_{1},...$ of points in $[0,1)^{s}$ is a $(t,s)$-sequence in base $b$ if, for all integers $k\geq 0$ and $m\geq t$, the point set consisting of ${\bf x}_{n}$ with $kb^{m}\leq n<(k+1)b^{m}$ is a $(t,m,s)$-net in base $b$.*

By [Ni, p. 56,60], $(t,m,s)$ nets and $(t,s)$ sequences are of low discrepancy. See reviews on $(t,m,s)$ nets and $(t,s)$ sequences in [DiPi] and [Ni].

Definition 4. ([DiPi, Definition 4.30]) *For a given dimension $s\geq 1$, an integer base $b\geq 2$, and a function ${\bf T}:\mathbb{N}_{0}\to\mathbb{N}_{0}$ with ${\bf T}(m)\leq m$ for all $m\in\mathbb{N}_{0}$, a sequence $({\bf x}_{0},{\bf x}_{1},...)$ of points in $[0,1)^{s}$ is called a $({\bf T},s)$-sequence in base $b$ if for all integers $m\geq 0$ and $k\geq 0$, the point set consisting of the points $x_{kb^{m}},...,x_{kb^{m}+b^{m}-1}$ forms a $({\bf T}(m),m,s)$-net in base $b$.

*Definition 5. ([DiPi, Definition 4.47]) Let $m,s\geq 1$ be integers. Let $C^{(1,m)},...,C^{(s,m)}$ be $m\times m$ matrices over $\mathbb{F}_{b}$. Now we construct $b^{m}$ points in $[0,1)^{s}$. For $n=0,1,...,b^{m}-1$, let $n=\sum^{m-1}_{j=0}a_{j}(n)b^{j}$ be the $b$-adic expansion of $n$. Choose a bijection $\phi:\;\mathbb{Z}_{b}:=\{0,1,....,b-1\}\mapsto\mathbb{F}_{b}$ with $\phi(0)=\bar{0}$, the neutral element of addition in $\mathbb{F}_{b}$. We identify $n$ with the row vector

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We map the vectors

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to the real numbers

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to obtain the point

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The point set $\{{\bf x}_{0},...,{\bf x}_{b^{m}-1}\}$ is called a digital net (over $\mathbb{F}_{b}$) (with generating matrices $(C^{(1,m)},...,C^{(s,m)})$).

*For $m=\infty$, we obtain a sequence ${\bf x}_{0},{\bf x}_{1},...$ of points in $[0,1)^{s}$ which is called a digital sequence (over $\mathbb{F}_{b}$) (with generating matrices $(C^{(1,\infty)},...,C^{(s,\infty)})$).

*We abbreviate $C^{(i,m)}$ as $C^{(i)}$ for $m\in\mathbb{N}$ and for $m=\infty$.

2.2 Duality theory (see [DiPi, Section 7]).

Let $\mathcal{N}$ be an arbitrary $\mathbb{F}_{b}$-linear subspace of $\mathbb{F}_{b}^{sm}$. Let $H$ be a matrix over $\mathbb{F}_{b}$ consisting of $sm$ columns such that the row-space of $H$ is equal to $\mathcal{N}$. Then we define the dual space $\mathcal{N}^{\bot}\subseteq\mathbb{F}_{b}^{sm}$ of $\mathcal{N}$ to be the null space of $H$ (see [DiPi, p. 244]). In other words, $\mathcal{N}^{\bot}$ is the orthogonal complement of $\mathcal{N}$ relative to the standard inner product in $\mathbb{F}_{b}^{sm}$,

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Let $C^{(1)},...,C^{(s)}\in\mathbb{F}_{b}^{\infty\times\infty}$ be generating matrices of a digital sequence $({\bf x}_{n}(C))_{n\geq 0}$ over $\mathbb{F}_{b}$. For any $m\in\mathbb{N}$, we denote the $m\times m$ left-upper sub-matrix of $C^{(i)}$ by $[C^{(i)}]_{m}$. The matrices $[C^{(1)}]_{m},...,[C^{(s)}]_{m}$ are then the generating matrices of a digital net. We define the overall generating matrix of this digital net by

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for any $m\in\mathbb{N}$.

Let $\mathcal{C}_{m}$ denote the row space of the matrix $[C]_{m}$ i.e.,

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The dual space is then given by

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Lemma A. ([DiPi, Theorem 4.86]) *Let $b$ be a prime power. A strict digital $({\bf T},s)$-sequence over $\mathbb{F}_{b}$ is uniformly distributed modulo one, if and only if $\liminf_{m\to\infty}(m-{\bf T}(m))=\infty$.

*2.3 Admissible sequences.

For $x=\sum_{j\geq 1}x_{j}b^{-j}$, and $y=\sum_{j\geq 1}y_{j}b^{-j}$ where $x_{j},y_{j}\in\mathbb{Z}_{b}:=\{0,1,....,b\\ -1\}$, we define the ($b$-adic) digital shifted point $v$ by $v=x\oplus y:=\sum_{j\geq 1}v_{j}b^{-j}$, where $v_{j}\equiv x_{j}+y_{j}\;({\rm mod}\;b)$ and $v_{j}\in\mathbb{Z}_{b}$. Let ${\bf x}=(x^{(1)},...,x^{(s)})\in[0,1)^{s}$, ${\bf y}=(y^{(1)},...,y^{(s)})\in[0,1)^{s}$. We define the ($b$-adic) digital shifted point ${\bf v}$ by ${\bf v}={\bf x}\oplus{\bf y}=(x^{(1)}\oplus y^{(1)},...,x^{(s)}\oplus y^{(s)})$. For $n_{1},n_{2}\in[0,b^{m})$, we define $n_{1}\oplus n_{2}:\\ =(n_{1}/b^{m}\oplus n_{2})b^{m})b^{m}$.

For $x=\sum_{j\geq 1}x_{i}b^{-i}$, where $x_{i}\in\mathbb{Z}_{b}$, $x_{i}=0$ $(i=1,...,k)$ and $x_{k+1}\neq 0$, we define the absolute valuation $\left\|.\right\|_{b}$ of $x$ by $\left\|x\right\|_{b}=b^{-k-1}$. Let $\left\|n\right\|_{b}=b^{k}$ for $n\in[b^{k},b^{k+1})$.

Definition 6. A point set $({\bf x}_{n})_{0\leq n<b^{m}}$ in $[0,1)^{s}$ is $d-$admissible in base $b$ if

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*A sequence $({\bf x}_{n})_{n\geq 0}$ in $[0,1)^{s}$ is $d-$admissible in base $b$ if * $\inf_{n>k\geq 0}\left\|n\ominus k\right\|_{b}\\ \times\left\|{\bf x}_{n}\ominus{\bf x}_{k}\right\|_{b}\geq b^{-d}$.

By [Le2], generalized Niederreiter’s sequences, Xing-Niederreiter’s sequences and Halton-type $(t,s)$ sequences have $d-$admissible properties. In [Le2], we proved for all $d-$admissible digital $(t,s)$ sequences $({\bf x}_{n})_{n\geq 0}$

[TABLE]

with some ${\bf w}$ and $K>0$. This result supports conjecture (1.1).

Definition 7. A sequence $({\bf x}_{n})_{n\geq 0}$ in $[0,1)^{s}$ is weakly admissible in base $b$ if

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Let $m\geq 1$, $\tau_{m}=[\log_{q}(\kappa_{m})]+m$, ${\bf w}=(w^{(1)},...,w^{(s)})$, $w^{(i)}=(w^{(i)}_{1},...,w^{(i)}_{\tau_{m}})$,

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Theorem B. (see [Le3, Proposition]) *Let $({\bf x}_{n})_{n\geq 0}$ be a uniformly distributed weakly admissible digital $(T,s)$-sequence in base $b$, satisfying (2.6) for all $m\geq m_{0}$. Then the set $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is of bounded remainder with respect to $({\bf x}_{n})_{n\geq 0}$ if and only if (1.2) is true.

2.4 Notation and terminology for algebraic function fields. For the theory of algebraic function fields, we follow the notation and terminology in the books [St] and [Sa].

Let $b$ be an arbitrary prime power, $\mathbb{F}_{b}$ a finite field with $b$ elements, $\mathbb{F}_{b}(x)$ the rational function field over $\mathbb{F}_{b}$, and $\mathbb{F}_{b}[x]$ the polynomial ring over $\mathbb{F}_{b}$. For $\alpha=f/g,\;f,g\in\mathbb{F}_{b}[x]$, let

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be the degree valuation of $\mathbb{F}_{b}(x)$. We define the field of Laurent series as

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A finite extension field $F$ of $\mathbb{F}_{b}(x)$ is called an algebraic function field over $\mathbb{F}_{b}$. Let $\mathbb{F}_{b}$ be algebraically closed in $F$. We express this fact by simply saying that $F/\mathbb{F}_{b}$ is an algebraic function field. The genus of $F/\mathbb{F}_{b}$ is denoted by $g$.

A place $\mathcal{P}$ of $F$ is, by definition, the maximal ideal of some valuation ring of $F$. We denote by $O_{\mathcal{P}}$ the valuation ring corresponding to $\mathcal{P}$ and we denote by $\mathbb{P}_{F}$ the set of places of $F$. For a place $\mathcal{P}$ of $F$, we write $\nu_{\mathcal{P}}$ for the normalized discrete valuation of $F$ corresponding to $\mathcal{P}$, and any element $t\in F$ with $\nu_{\mathcal{P}}(t)=1$ is called a local parameter (prime element) at $\mathcal{P}$.

The field $F_{\mathcal{P}}:=O_{\mathcal{P}}/\mathcal{P}$ is called the residue field of $F$ with respect to $\mathcal{P}$. The degree of a place $\mathcal{P}$ is defined as ${\rm deg}(\mathcal{P})=[F_{\mathcal{P}}:\mathbb{F}_{b}]$. We denote by ${\rm Div}(F)$ the set of divisors of $F/\mathbb{F}_{b}$.

The completion of $F$ with respect to $\nu_{\mathcal{P}}$ will be denoted by $F^{(\mathcal{P})}$. Let $t$ be a local parameter of $\mathcal{P}$. Then $F^{(\mathcal{P})}$ is isomorphic to $F_{\mathcal{P}}((t))$ (see [Sa, Theorem 2.5.20]), and an arbitrary element $\alpha\in F^{(P)}$ can be uniquely expanded as (see [Sa, p. 293])

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The derivative $\frac{{\rm d}\alpha}{{\rm d}t}$, or differentiation with respect to $t$, is defined by (see [Sa, Definition 9.3.1])

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For an algebraic function field $F/\mathbb{F}_{b}$, we define its set of differentials (or Hasse differentials, H-differentials) as

[TABLE]

(see [St, Definition 4.1.7]).

Lemma B. ([St, Proposition 4.1.8] or [Sa, Theorem 9.3.13]) Let $z\in F$ be separating. Then every differential $\gamma\in\Delta_{F}$ can be written uniquely as $\gamma=y\;{\rm d}z$ for some $y\in F$.

We define the order of $\alpha\;{\rm d}\beta$ at $\mathcal{P}$ by

[TABLE]

where $t$ is any local parameter for $\mathcal{P}$ (see [Sa, Definition 9.3.8]).

Let $\Omega_{F}$ be the set of all Weil differentials of $F/\mathbb{F}_{b}$. There exists an $F-$linear isomorphism of the differential module $\Delta_{F}$ onto $\Omega_{F}$ (see [St, Theorem 4.3.2] or [Sa, Theorem 9.3.15]).

For $0\neq\omega\in\Omega_{F}$, there exists a uniquely determined divisor ${\rm div}(\omega)\in{\rm Div}(F)$. Such a divisor ${\rm div}(\omega)$ is called a canonical divisor of $F/\mathbb{F}_{b}$. (see [St, Definition 1.5.11]). For a canonical divisor $\dot{W}$, we have (see [St, Corollary 1.5.16])

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Let $\alpha\;{\rm d}\beta$ be a nonzero H-differential in $F$ and let $\omega$ be the corresponding Weil differential. Then (see [Sa, Theorem 9.3.17], [St, ref. 4.35])

[TABLE]

Let $\alpha\;{\rm d}\beta$ be an H-differential, $t$ a local parameter of $\mathcal{P}$, and

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Then the residue of $\alpha\;{\rm d}\beta$ (see [Sa, Definition 9.3.10) is defined by

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Let

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For a divisor $\mathcal{D}$ of $F/\mathbb{F}_{b}$, let $\mathcal{L}(\mathcal{D})$ denote the Riemann-Roch space

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Then $\mathcal{L}(\mathcal{D})$ is a finite-dimensional vector space over $\mathbb{F}$, and we denote its dimension by $\ell(\mathcal{D})$. By [St, Corollary 1.4.12],

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Theorem C (Riemann-Roch Theorem). [St, Theorem 1.5.15, and St, Theorem 1.5.17 ] *Let $W$ be a canonical divisor of $F/\mathbb{F}_{b}$. Then for each divisor $A\in{\rm div}(F)$, $\ell(A)={\rm deg}(A)+1-g+\ell(W-A)$, and *

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3 Statements of results.

3.1 Generalized Niederreiter sequence. In this subsection, we introduce a generalization of the Niederreiter sequence due to Tezuka (see [DiPi, Section 8.1.2]). By [DiPi, Section 8.1], the Sobol’s sequence, the Faure’s sequence and the original Niederreiter sequence are particular cases of a generalized Niederreiter sequence.

Let $b$ be a prime power and let $p_{1},...,p_{s}\in\mathbb{F}_{b}[x]$ be pairwise coprime polynomials over $\mathbb{F}_{b}$. Let $e_{i}={\rm deg}(p_{i})\geq 1$ for $1\leq i\leq s$. For each $j\geq 1$ and $1\leq i\leq s$, the set of polynomials $\{y_{i,j,k}(x)\;:\;0\leq k<e_{i}\}$ needs to be linearly independent $({\rm mod}\;p_{i}(x))$ over $\mathbb{F}_{b}$. For integers $1\leq i\leq s$, $j\geq 1$ and $0\leq k<e_{i}$, consider the expansions

[TABLE]

over the field of formal Laurent series $\mathbb{F}_{b}((x^{-1}))$. Then we define the matrix $C^{(i)}=(c^{(i)}_{j,r})_{j\geq 1,r\geq 0}$ by

[TABLE]

where $j-1=Qe_{i}+k$ with integers $Q=Q(i,j)$ and $k=k(i,j)$ satisfying $0\leq k<e_{i}$.

A digital sequence $({\bf x}_{n})_{n\geq 0}$ over $\mathbb{F}_{b}$ generated by the matrices $C^{(1)},...,C^{(s)}$ is called a generalized Niederreiter sequence (see [DiPi, p.266]).

Theorem D. (see [DiPi, p.266] and [Le1, Theorem 1]) The generalized Niederreiter sequence $({\bf x}_{n})_{n\geq 0}$ with generating matrices, defined as above, is a digital $d-$admissible (t, s)-sequence over $\mathbb{F}_{b}$ with $d=e_{0}$, $t=e_{0}-s$ and $e_{0}=e_{1}+...+e_{s}.$

In this paper, we will consider the case where $(x,p_{i})=1$ for $1\leq i\leq s$. We will consider the general case in [Le4].

Theorem 1. With the notations as above, the set $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is of bounded remainder with respect to $({\bf x}_{n})_{n\geq 0}$ if and only if (1.2) is true.

3.2 Xing-Niederreiter sequence (see [DiPi, Section 8.4 ]). Let $F/\mathbb{F}_{b}$ be an algebraic function field with full constant field $\mathbb{F}_{b}$ and genus $g$. Assume that $F/\mathbb{F}_{b}$ has at least one rational place $P_{\infty}$, and let $G$ be a positive divisor of $F/\mathbb{F}_{b}$ with ${\rm deg}(G)=2g$ and $P_{\infty}\notin{\rm supp}(G)$. Let $P_{1},...,P_{s}$ be $s$ distinct places of $F/\mathbb{F}_{b}$ with $P_{i}\neq P_{\infty}$ for $1\leq i\leq s$. Put $e_{i}={\rm deg}(P_{i})$ for $1\leq i\leq s$.

By [DiPi, p.279 ], we have that there exists a basis ${w_{0},w_{1},...,w_{g}}$ of $\mathcal{L}(G)$ over $\mathbb{F}_{b}$ such that

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where $0=n_{0}<n_{1}<....<n_{g}\leq 2g$. For each $1\leq i\leq s$, we consider the chain

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of vector spaces over $\mathbb{F}_{b}$. By starting from the basis ${w_{0},w_{1},...,w_{g}}$ of $\mathcal{L}(G)$ and successively adding basis vectors at each step of the chain, we obtain for each $n\in\mathbb{N}$ a basis

[TABLE]

of $\mathcal{L}(G+nP_{i})$. We note that we then have

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Lemma C. ([DiPi, Lemma 8.10]) *The system $\{w_{0},w_{1},...,w_{g}\}\cup\{k^{(i)}_{j}\}_{1\leq i\leq s,j\geq 1}$ of elements of $F$ is linearly independent over $\mathbb{F}_{b}$. *

Let $z$ be an arbitrary local parameter at $P_{\infty}$. For $r\in\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$, we put

[TABLE]

Note that in this case $\nu_{P_{\infty}}(z_{r})=r$ for all $r\in\mathbb{N}_{0}$. For $1\leq i\leq s$ and $j\in\mathbb{N}$, we have $k^{(i)}_{j}\in\mathcal{L}(G+nP_{i})$ for some $n\in\mathbb{N}$ and also $P_{\infty}\notin{\rm supp}(G+nP_{i})$, hence $\nu_{P_{\infty}}(k_{j}^{(i)})\geq 0$. Thus we have the local expansions

[TABLE]

where all coefficients $a_{j,r}^{(i)}\in\mathbb{F}_{b}$. Let $H_{1}=\mathbb{N}_{0}\setminus H_{2}=\{h(0),h(1),...\}$, $H_{2}\\ =\{n_{0},n_{1},...,n_{g}\}$.

For $1\leq i\leq s$ and $j\in\mathbb{N}$, we now define the sequences

[TABLE]

[TABLE]

where the hat indicates that the corresponding term is deleted. We define the matrices $C^{(1)},...,C^{(s)}\in\mathbb{F}^{\mathbb{N}\times\mathbb{N}}_{b}$ by

[TABLE]

i.e., the vector ${\bf c}^{(i)}_{j}$ is the $j$th row vector of $C^{(i)}$ for $1\leq i\leq s$.

Theorem E (see [DiPi, Theorem 8.11] and [Le1, Theorem 1]). *With the above notations, we have that the matrices $C^{(1)},...,C^{(s)}$ given by (3.5) are generating matrices of the Xing-Niederreiter $d-$admissible digital $(t,s)$-sequ-

ence $({\bf x}_{n})_{n\geq 0}$ with $d=e_{1}+...+e_{s}$, $t=g+e_{1}+...+e_{s}-s$.*

In order to obtain the bounded remainder set property, we will take a specific local parameter $z$. Let $P_{0}\in\mathbb{P}_{F}$, $P_{0}\not\subset\{P_{1},...,P_{s},P_{\infty}\}$, $P_{0}\notin{\rm supp}(G)$ and ${\rm deg}(P_{0})=e_{0}$. By the Riemann-Roch theorem, there exists a local parameter $z$ at $P_{\infty}$, with

[TABLE]

Theorem 2. With the notations as above, the set $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is of bounded remainder with respect to $({\bf x}_{n})_{n\geq 0}$ if and only if (1.2) is true.

3.3 Generalized Halton-type sequences from global function fields.

Let $q\geq 2$ be an integer

[TABLE]

Van der Corput proved that $(\varphi_{q}(n))_{n\geq 0}$ is a $1-$dimensional l.d.s. Let

[TABLE]

where $\hat{q}_{1},\ldots,\hat{q}_{s}\geq 2$ are pairwise coprime integers. Halton proved that $(\hat{H}_{s}(n))_{n\geq 0}$ is an $s-$dimensional l.d.s. (see [Ni]).

Let $Q=(q_{1},q_{2},....)$ and $Q_{j}=q_{1}q_{2}....q_{j}$, where $q_{j}\geq 2$  $(j=1,2,\dots)$ is a sequence of integers. Every nonnegative integer $n$ then has a unique $Q$-adic representation of the form

[TABLE]

where $n_{j}\in\{0,1,...,q_{j}-1\}$. We call this the Cantor expansion of $n$ with respect to the base $Q$. Consider Cantor’s expansion of $x\in[0,1):$

[TABLE]

The $Q-$adic representation of $x$ is then unique. We define the radical inverse function

[TABLE]

Let $p_{i,j}\geq 2$ be integers $(s\geq i\geq 1,j\geq 1)$, $g.c.d.(p_{i,k},p_{j,l})=1$ for $i\neq j$, $\tilde{P}_{i,0}=1,\;\;\tilde{P}_{i,j}=\prod_{1\leq k\leq j}p_{i,k},\;i\in[1,s],\;j\geq 1$, $\mathcal{P}_{i}=(p_{i,1},p_{i,2},...)$, $\boldsymbol{\mathcal{P}}=(\mathcal{P}_{1},...,\mathcal{P}_{s})$. In [He], Hellecaleq proposed the following generalisation of the Halton sequence:

[TABLE]

In [Te], Tezuka introduced a polynomial arithmetic analogue of the Halton sequence :

Let $p(x)$ be an arbitrary nonconstant polynomial over $\mathbb{F}_{b}$, $e={\rm deg}(p)$, $n=a_{0}(n)+a_{1}(n)b+\cdots+a_{m}(n)b^{m}$. We fix a bijection $\phi\;:\;\mathbb{Z}_{b}\to\mathbb{F}_{b}$ with $\phi(0)=\bar{0}$. Denote $v_{n}(x)=\bar{a}_{0}(n)+\bar{a}_{1}(n)x+\cdots+\bar{a}_{m}(n)x^{m}$, where $\bar{a}_{r}(n)=\phi(a_{r}(n))$, $r=0,1,...,m$. Then $v_{n}(x)$ can be represented in terms of $p(x)$ in the following way:

[TABLE]

We define the radical inverse function $\varphi_{p(x)}:\mathbb{F}_{b}[x]\to\mathbb{F}_{b}(x)$ as follows

[TABLE]

Let $p_{1}(x),...,p_{s}(x)$ be pairwise coprime. Then Tezuka’s sequence is defined as follows

[TABLE]

where each $\sigma_{i}$ is a mapping from $F$ to the real field defined by $\sigma_{i}(\sum_{j\geq w}\dot{a}_{j}x^{-j})\\ =\sum_{j\geq w}\phi^{-1}(\dot{a}_{j})b^{-j}$. By [Te], $({\bf x}_{n})_{n\geq 0}$ is a $(t,s)$ sequence in base $b$.

In 2010, Levin [Le1] and in 2013, Niederreiter and Yeo [NiYe] generalized Tezuka’s construction to the case of arbitrary algebraic function fields $F$. The construction of [NiYe] is follows:

Let $F/\mathbb{F}_{b}$ be an algebraic function field with full constant field $\mathbb{F}_{b}$ and genus $g$. We assume that $F/\mathbb{F}_{b}$ has at least one rational place, that is, a place of degree 1. Given a dimension $s\geq 1$, we choose $s+1$ distinct places $P_{1}$,…,$P_{s}$, $P_{\infty}$ of $F$ with deg$(P_{\infty})=1$. The degrees of the places $P_{1}$,…,$P_{s}$ are arbitrary and we put $e_{i}={\rm deg}(P_{i})$ for $1\leq i\leq s$. Denote by $O_{F}$ the holomorphy ring given by $O_{F}=\bigcap_{P\neq P_{\infty}}O_{P}$, where the intersection is extended over all places $P\neq P_{\infty}$ of $F$, and $O_{P}$ is the valuation ring of $P$. We arrange the elements of $O_{F}$ into a sequence by using the fact that $O_{F}=\bigcup_{m\geq 0}\mathcal{L}(mP_{\infty})$. The terms of this sequence are denoted by $f_{0},f_{1},...$ and they are obtained as follows. Consider the chain $\mathcal{L}(0)\subseteq L(P_{\infty})\subseteq L(2P_{\infty})\subseteq\cdots$ of vector spaces over $\mathbb{F}_{b}$. At each step of this chain, the dimension either remains the same or increases by $1$. From a certain point on, the dimension always increases by $1$ according to the Riemann-Roch theorem. Thus we can construct a sequence $v_{0},v_{1},...$ of elements of $O_{F}$ such that $\{v_{0},v_{1},...,v_{\ell(mP_{s+1})-1}\}$ is a $\mathbb{F}_{b}$-basis of $\mathcal{L}(mP_{s+1})$. We fix a bijection $\phi\;:\;\mathbb{Z}_{b}\to\mathbb{F}_{b}$ with $\phi(0)=\bar{0}$. Then we define

[TABLE]

Note that the sum above is finite since for each $n\in\mathbb{N}$. We have $a_{r}(n)=0$ for all sufficiently large $r$. By the Riemann-Roch theorem, we have

[TABLE]

For each $i=1,...,s$, let $\wp_{i}$ be the maximal ideal of $O_{F}$ corresponding to $P_{i}$. Then the residue class field $F_{P_{i}}:=O_{F}/\wp_{i}$ has order $b^{e_{i}}$ (see [St, Proposition 3.2.9]). We fix a bijection $\sigma_{P_{i}}\;:\;F_{P_{i}}\to Z_{b^{e_{i}}}$. For each $i=1,...,s$, we can obtain a local parameter $t_{i}\in O_{F}$ at $\wp_{i}$, by applying the Riemann-Roch theorem and choosing $t_{i}\in\mathcal{L}(kP_{\infty}-P_{i})\setminus\mathcal{L}(kP_{\infty}-2P_{i})$ for a suitably large integer $k$. We have a local expansion of $f_{n}$ at $\wp_{i}$ of the form

[TABLE]

We define the map $\xi\;:\;O_{F}\to[0,1]^{s}$ by

[TABLE]

Now we define the sequence ${\bf x}_{0},{\bf x}_{1},...$ of points in $[0,1]^{s}$ by ${\bf x}_{n}=\xi(f_{n})\;\;\;\;{\rm for}$

$n=0,1,...$ . From [NiYe, Theorem 1], we get the following theorem :

Theorem F. *With the notation as above, we have that $({\bf x}_{n})_{n\geq 0}$ is a $(t,s)$-sequence over $\mathbb{F}_{b}$ with $t=g+e_{1}+...+e_{s}-s$.

The construction of Levin [Le1] is similar, but more complicated than in [NiYe]. However in [Le1], we can use arbitrary pairwise coprime divisors $D_{1},...,D_{s}$ instead of places $P_{1},...,P_{s}$.

In this paper, we introduce the Hellecalek-like generalisation (3.7) of the above construction:

Let $\mathbb{P}_{F}:=\{P|P\;{\rm be\;a\;place\;of\;}F/\mathbb{F}_{b}\}$, $P_{0},P_{\infty}\in\mathbb{P}_{F}$, ${\rm deg}(P_{\infty})=1$, ${\rm deg}(P_{0})\\ =e_{0}$, $P_{0}\neq P_{\infty}$, $P_{i,j}\in\mathbb{P}_{F}$ for $1\leq j,1\leq i\leq s$, $P_{i_{1},j_{1}}\neq P_{i_{2},j_{2}}$ for $i_{1}\neq i_{2}$, $P_{i,j}\neq P_{0}$, $P_{i,j}\neq P_{\infty}$ for all $i,j$, $\dot{n}_{i,j}={\rm deg}(P_{i,j})$, $n_{i,j}={\rm deg}(\mathcal{P}_{i,j}),\;\mathcal{P}_{0,j}=P_{0}^{j}$,

[TABLE]

Let $i\in[0,s]$. We will construct a basis $(w^{(i)}_{j})_{j\geq 0}$ of $O_{F}$ in the following way. Let

[TABLE]

Using the Riemann-Roch theorem, we obtain

[TABLE]

Let $(u^{(i)}_{j,\mu})_{\mu=1}^{g}$ be a $\mathbb{F}_{b}$ linear basis of $\mathfrak{L}_{i,j}$. By (3.9) and (3.10), we get that the basis $(u^{(i)}_{j,\mu})_{\mu=1}^{g}$ can be extended to a basis $(v^{(i)}_{j,1},\cdots,v^{(i)}_{j,\dot{n}_{i,j}},u^{(i)}_{j,1},\cdots,u^{(i)}_{j,g})$ of $\mathscr{L}_{i,j}$.

Bearing in mind that $(u^{(i)}_{j,\mu})_{\mu=1}^{g}$ is a $\mathbb{F}_{b}$ linear basis of $\mathfrak{L}_{i,j}$, we obtain that $v^{(i)}_{j,\mu}\notin\mathfrak{L}_{i,j}$ for $\mu\in[1,\dot{n}_{i,j}]$. So

[TABLE]

Let

[TABLE]

We claim that vectors from $V_{i,j}$ are $\mathbb{F}_{b}$ linear independent. Suppose the opposite. Assume that there exists $b^{(i)}_{k,\mu}\in\mathbb{F}_{b}$ such that

[TABLE]

Let $w_{l}\neq 0$ for some $l\in[1,j]$ and let $w_{k}=0$ for all $k\in[1,l)$.

Using (3.9) - (3.11), we get

[TABLE]

Applying definition (2.11) of the Riemann-Roch space, we obtain

[TABLE]

But from (3.13), (3.8) and (3.11), we have

[TABLE]

We have a contradiction. Hence vectors from $V_{i,j}$ are $\mathbb{F}_{b}$ linear independent.

By (3.9) - (3.12), we have $V_{i,j}\subset L_{i,j}$ and

[TABLE]

Hence vectors from $V_{i,j}$ are the $\mathbb{F}_{b}$ linear basis of $L_{i,j}$.

Now we will find a basis of $L_{i,j-2g}$. We claim that $u^{(i)}_{j,\mu}\notin L_{i,j-2g}$ for $\mu\in[1,g]$. Suppose the opposite. By (3.9) and (3.10), we get

[TABLE]

By (3.8), ${\rm deg}(T)=n_{i,j-2g}+2g-1-n_{i,j}<0$. Hence $\mathcal{L}(T)=\{0\}$. We have a contradiction. Bearing in mind that $V_{i,j}$ is $\mathbb{F}_{b}$ linear basis of $L_{i,j}$, we obtain that a basis of $L_{i,j-2g}$ can be chosen from the set $(v^{(i)}_{1,1},...,v^{(i)}_{1,\dot{n}_{i,1}},...,v^{(i)}_{j,1},...,v^{(i)}_{j,\dot{n}_{i,j}})$

$=V_{i,j}\setminus\{u^{(i)}_{j,\mu}\;|\;\mu=1,...,g\}$. From (3.9) - (3.11), we get

[TABLE]

Hence vectors

[TABLE]

$1\leq\mu\leq g$, for some $\rho\in[1,\dot{n}_{i,k}]$ and $k\in(j-2g,j]$ are an $\mathbb{F}_{b}$ linear basis of $L_{i,j-2g}$ $(0\leq i\leq s)$.

Therefore $(v^{(i)}_{k,\mu})_{1\leq\mu\leq\dot{n}_{i,k,k\geq 1}}$ is the $\mathbb{F}_{b}$ linear basis of $O_{F}=\cup_{j\geq 1}L_{i,j}$. We put in order the basis $(v^{(i)}_{k,\mu})_{1\leq\mu\leq\dot{n}_{i,k},k\geq 1}$ as follows

[TABLE]

So we proved the following lemma :

Lemma 1. For all $i\in[0,s]$ there exists a sequence $(w^{(i)}_{j})_{j\geq 0}$ such that $(w^{(i)}_{j})_{j\geq 0}$ is a $\mathbb{F}_{b}$ linear basis of $O_{F}$ and for all $j\geq 1$ a $\mathbb{F}_{b}$ linear basis of $L_{i,j}$ can be chosen from the set $\{w^{(i)}_{0},...,w^{(i)}_{n_{i,j+2g}-1}\}$.

Bearing in mind that $(w^{(i)}_{j})_{j\geq 0}$ is the $\mathbb{F}_{b}$ linear basis of $O_{F}$, we obtain for all $i\in[1,s]$ and $r\geq 0$ that there exists $c^{(i)}_{j,r}\in\mathbb{F}_{b}$ and integers $l^{(i)}_{r}$ such that

[TABLE]

Let $n=\sum_{r\geq 0}a_{r}(n)b^{r}$. We fix a bijection $\phi\;:\;\mathbb{Z}_{b}\to\mathbb{F}_{b}$ with $\phi(0)=\bar{0}$. Then we define

[TABLE]

By (3.15), we have for $i\in[0,s]$

[TABLE]

where $y^{(i)}_{n,j}=\sum_{r\geq 0}\bar{a}_{r}(n)c^{(i)}_{j,r}\in\mathbb{F}_{b}$, $y^{(0)}_{n,j}=\bar{a}_{j-1}(n)$.

We map the vectors

[TABLE]

to the real numbers

[TABLE]

to obtain the point

[TABLE]

Theorem 3. With the notations as above, the set $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is of bounded remainder with respect to $({\bf x}_{n})_{n\geq 0}$ if and only if (1.2) is true.

Remark. It is easy to verify that Hellekalek’s sequence and our generalized Halton-type sequence $({\bf x}_{n})_{n\geq 0}$ are l.d.s if

[TABLE]

3.4 Niederreiter-Xing sequence (see [DiPi, Section 8.3 ]). Let $F/\mathbb{F}_{b}$ be an algebraic function field with full constant field $\mathbb{F}_{b}$ and genus $g$. Assume that $F/\mathbb{F}_{b}$ has at least $s+1$ rational places. Let $P_{1},...,P_{s+1}$ be $s+1$ distinct rational places of $F$. Let $G_{m}=m(P_{1}+...+P_{s})-(m-g+1)P_{s+1}$, and let $t_{i}$ be a local parameter at $P_{i}$, $1\leq i\leq s+1$. For any $f\in\mathcal{L}(G_{m})$ we have $\nu_{P_{i}}(f)\geq-m$, and so the local expansion of $f$ at $P_{i}$ has the form

[TABLE]

For $1\leq i\leq s$, we define the $\mathbb{F}_{b}$-linear map $\psi_{m,i}\;:\;\mathcal{L}(G_{m})\to\mathbb{F}^{m}_{b}$ by

[TABLE]

Let

[TABLE]

Let $C^{(1)},...,C^{(s)}\in\mathbb{F}_{b}^{\infty\times\infty}$ be the generating matrices of a digital sequence ${\bf x}_{n}(C)_{n\geq 0}$, and let $(\mathcal{C}_{m})_{m\geq 1}$ be the associated sequence of row spaces of overall generating matrices $[C]_{m}$, $m=1,2,...$ (see (2.5)).

Theorem G. (see [DiPi, Theorem 7.26 and Theorem 8.9]) There exist matrices $C^{(1)},...,C^{(s)}$ such that $({\bf x}_{n}(C))_{n\geq 0}$ is a digital $(t,s)$-sequence with $t=g$ and $\mathcal{C}_{m}^{\bot}=\mathcal{M}_{m}(P_{1},...,P_{s};G_{m})$ for $m\geq g+1$, $s\geq 2$.

In [Le2, p.24], we proposed the following way to get ${\bf x}_{n}(C)_{n\geq 0}$ :

We consider the $H$-differential $dt_{s+1}$. Let $\omega$ be the corresponding Weil differential, ${\rm div}(\omega)$ the divisor of $\omega$, and $W:={\rm div}(dt_{s+1})={\rm div}(\omega)$. By (2.7)-(2.9), we have ${\rm deg}(W)=2g-2$. We consider a sequence $\dot{v}_{0},\dot{v}_{1},...$ of elements of $F$ such that $\{\dot{v}_{0},\dot{v}_{1},...,\dot{v}_{\ell((m-g+1)P_{s+1}+W)-1}\}$ is an $\mathbb{F}_{b}$ linear basis of $L_{m}:=\mathcal{L}((m-g+1)P_{s+1}+W)$ and

[TABLE]

for $2\leq j<r+2-g$, where

[TABLE]

According to Lemma B, we have that there exists $\tau_{i}\in F$ $(1\leq i\leq s)$ such that ${\rm d}t_{s+1}=\tau_{i}{\rm d}t_{i},\quad{\rm for}\quad 1\leq i\leq s$.

Bearing in mind (2.8), (2.10) and (3.20), we get

[TABLE]

We consider the following local expansions

[TABLE]

Now let $\dot{C}^{(i)}=(\dot{c}^{(i)}_{j,r})_{j-1,r\geq 0}$, $1\leq i\leq s$, and let $(\dot{\mathcal{C}}_{m}^{\bot})_{m\geq 1}$ be the associated sequence of row spaces of overall generating matrices $[\dot{C}]_{m}$, $m=1,2,...$ (see (2.5)).

Theorem H (see [Le2, Theorem 5]). With the above notations, $({\bf x}_{n}(\dot{C}))_{n\geq 0}$ is a digital $d-$admissible $(t,s)$ sequence with $d=g+s$, $t=g$, and $\dot{\mathcal{C}}_{m}^{\bot}=\mathcal{M}_{m}(P_{1},...,P_{s};G_{m})$ for all $m\geq g+1$.

We note that condition (3.20) is required in the proof of Theorem H only in order to get the discrepancy lower bound. While the equality $\dot{\mathcal{C}}_{m}^{\bot}=\mathcal{M}_{m}(P_{1},...,P_{s};G_{m})$ is true for arbitrary sequence $\dot{v}_{0},\dot{v}_{1},...$ of elements of $\mathbb{F}_{b}$ such that for all $m\geq 1$

[TABLE]

In order to obtain the bounded remainder property, in this paper, we will construct from $(\dot{v}_{n})_{n\geq 0}$ a special basis $(\ddot{v}_{n})_{n\geq 0}$ as follows :

Let $P_{0}\in\mathbb{P}_{F}$, $P_{0}\neq P_{i}$ $(i=1,...,s+1)$, and let $t_{0}$ be a local parameter of $P_{0}$. For simplicity, we suppose that ${\rm deg}(P_{0})=1$. Let

[TABLE]

It is easy to verify that

[TABLE]

Using the Riemann-Roch theorem, we have that there exists

[TABLE]

According to Lemma B, we have that there exists $\tau_{0}\in F$, such that ${\rm d}t_{s+1}=\tau_{0}{\rm d}t_{0}$.

Let $u\in L_{m}=\mathcal{L}((m-g+1)P_{s+1}+W)$ with $m\geq 0$. Bearing in mind (2.8), (2.10), (3.23)-(3.25) and the Riemann-Roch theorem, we get

[TABLE]

and

[TABLE]

We consider the sequence $(\dot{v}_{j})_{j\geq 0}$ (3.20). By (3.22), $(\dot{v}_{j})_{j=0}^{m-1}$ is an $\mathbb{F}_{b}$ linear basis of $L_{m}$. Let

[TABLE]

It is easy to verify that $\alpha(j)\neq\alpha(j)$ for $i\neq j$. We construct a sequence $(\ddot{v}_{j})_{j\geq 0}$ as follows :

[TABLE]

It is easy to see that $(\ddot{v}_{j})_{j\geq 0}$ satisfy the condition (3.22). Bearing in mind (3.26)-(3.28) and that $\ddot{v}_{j}\in L_{m}$ for $j<m$, we get

[TABLE]

Hence, for all $f\in L_{m}$, we have

[TABLE]

Taking into account (3.27) and (3.30), we obtain

[TABLE]

Suppose that $\alpha(j)>j+g$. By (3.22) - (3.24), $\ddot{v}_{j}\in L_{j+1}=\mathcal{L}((j-g+2)P_{s+1}+W)$. Hence $\ddot{v}_{j}\in\mathcal{L}(X)$, with $X=(j-g+2)P_{s+1}+W-(j+g+1)P_{0}$.

Bearing in mind that ${\rm deg}(P_{0})={\rm deg}(P_{s+1})=1$ and ${\rm deg}(W)=2g-2$, we get ${\rm deg}(X)=-1$. Therefore $\mathcal{L}(X)=\{0\}$ and we have a contradiction. Hence

[TABLE]

By (3.31), we have that for every integer $k\geq 0$ there exists $r\geq 0$ with $\alpha(r)=k$. Therefore the map $\alpha:\;\mathbb{N}_{0}\to\mathbb{N}_{0}$ is an isomorphism. Hence there exist integers $\beta(k)\geq 0$ such that

[TABLE]

From (3.32), we get for $j=\beta(k)$

[TABLE]

Let

[TABLE]

Taking $r=\beta(k)$, we get $\alpha(r)=k$ and

[TABLE]

Suppose $j\notin B_{j+g+1}$ for some $j$, then $j=\beta(j+g+l)$ for some $l\geq 1$. Using (3.34) with $k=j+g+l$, we obtain

[TABLE]

We have a contradiction. Hence

[TABLE]

We consider the local expansion (3.21), applied to $i=0$ :

[TABLE]

Let $(x_{n}^{(0)}(\dot{C}^{(0)}))_{n\geq 0}$ be the digital sequence generated by the matrix $\dot{C}^{(0)}$.

Now we consider the matrix $\tilde{C}^{(i)}=(\dot{c}^{(i)}_{j,r})_{j-1,r\geq 0}$, obtained from equation (3.21) and (3.37), where we take $\ddot{v}_{r}$ instead of $\dot{v}_{r}$ $(i=0,1,...,s)$. Using Theorem H, we obtain that $(x_{n}^{(0)}(\tilde{C}^{(0)}),{\bf x}_{n}(\tilde{C}))_{n\geq 0}$ is the digital $(t,s+1)$-sequence with $t=g$. Therefore we have proved the following lemma :

Lemma 2. *There exists a sequence $(\ddot{v}_{j})_{j\geq 0}$ such that $(x_{n}^{(0)}(\tilde{C}^{(0)}),{\bf x}_{n}(\tilde{C}))_{n\geq 0}$ is the digital $(t,s+1)$-sequence with $t=g$ and $\{0,1,...,m-1\}\subset B_{m+g}$.

*In §4.4, we will prove

Theorem 4. With the notations as above, the set $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is of bounded remainder with respect to $({\bf x}_{n}(\tilde{C}))_{n\geq 0}$ if and only if (1.2) is true.

4 Proof

Consider the following condition

[TABLE]

We will prove (4.1) for the generalized Halton sequence in §4.3. For other considered sequences, assertion (4.1) follows from Theorem D, Theorem E and Theorem H.

The sufficient part of all considered theorems follows from Definition 2 and (4.1). Therefore we need only consider the case of necessity.

4.1 Generalized Niederreiter sequence. Proof of Theorem 1.

From Theorem D, we have that $({\bf x}_{n})_{n\geq 0}$ is the uniformly distributed digital weakly admissible $(t,s)$-sequence in base $b$. By Theorem B, in order to prove Theorem 1, we need only to check condition (2.6). By ( Le2, p.26, ref 4.6 ), we get

[TABLE]

[TABLE]

We take $\dot{y}_{i,j,k}(x)=x^{m}y_{i,j,k}(x)$ instead of $y_{i,j,k}(x)$. Now using Theorem D, we obtain from (4.2), (2.2) - (2.4) that $(\dot{{\bf x}}_{n})_{0\leq n<b^{\dot{m}}}$ is a $(t,\dot{m},s)$ net for $\dot{m}=s\tau_{m}+t$ with $\dot{x}_{n,j}^{(i)}=\phi^{-1}(\dot{y}_{n,j}^{(i)})=x_{b^{m}n,j}^{(i)}$. Bearing in mind that $\dot{{\bf x}}_{n}={\bf x}_{b^{m}n}$, we obtain (2.6). Hence Theorem 1 is proved.

4.2 Xing-Niederreiter sequence. Proof of Theorem 2.

By Theorem B and Theorem E, in order to prove Theorem 2, we need only to check condition (2.6).

From (2.1) - (2.4), we get that in order to obtain (2.6), it suffices to prove that

[TABLE]

$m_{0}=2g+2+e_{1}+\cdots+e_{s}$, for all $u^{(i)}_{j}\in\mathbb{F}_{b}$.

Let

[TABLE]

Let $k^{(0)}_{j+1}=z_{h(j)}=z^{h(j)}$ for $j\in H_{1}$ with $H_{1}=\mathbb{N}_{0}\setminus H_{2}=\{h(0),h(1),...\}$, $H_{2}=\{n_{0},n_{1},...,n_{g}\}$. From (3.3), we have

[TABLE]

Let $c^{(0)}_{j,r}:=a^{(0)}_{j,h(r)}$. By (2.2), (2.3) and (3.6), we get

[TABLE]

So, we obtain a digital $s+1$-dimensional sequence $(x^{(0)}_{n},{\bf x}_{n})_{n\geq 0}$.

Let $n=\sum_{r=0}^{M-1}a_{r}(n)b^{r}$ and let

[TABLE]

By (2.2), (3.4) and (4.4), we get

[TABLE]

We fix $\tilde{n}\in\ddot{U}$. Let

[TABLE]

It is easy to verify that statement (4.3) follows from the next assertion

[TABLE]

Taking into account that $y^{(i)}_{n,j}=y^{(i)}_{\dot{n},j}+y^{(i)}_{\ddot{n},j}$, we get

[TABLE]

where $\dot{u}^{(i)}_{j}=\dot{u}^{(i)}_{j}-y^{(i)}_{\tilde{n},j}$.

According to (2.2), (3.4) and (4.4), in order to prove (4.6) , it suffices to show that the vectors

[TABLE]

$d_{i}=\tau_{m}$, $1\leq i\leq s$ and $d_{0}=m$, are linearly independent over $\mathbb{F}_{b}$.

To prove this statement, we closely follow [DiPi, p.282]. Suppose that we have

[TABLE]

for some $f^{(i)}_{j}\in\mathbb{F}_{b}$ with $\sum_{j=1}^{m}|\phi^{-1}(f^{(0)}_{j})|+\sum_{i=1}^{s}\sum_{j=1}^{\tau_{m}}|\phi^{-1}(f^{(i)}_{j})|>0$.

We put $f^{(0)}_{r}=0$ for $r>m$. Hence

[TABLE]

By (3.4) and (4.4), we obtain $c^{(i)}_{j,r}=a^{(i)}_{j,h(r)}$ for $1\leq i\leq s$ and $c^{(0)}_{j,r}=\delta(j-1\\ =h(r))$. Therefore

[TABLE]

${\rm for}\;r\in[0,M)$.

Now consider the element $\alpha\in\mathbb{F}_{b}$ given by $\alpha=\alpha_{1}+\alpha_{2}$, where

[TABLE]

Using (3.3), we get

[TABLE]

From (3.4), (4.8) and (4.9), we obtain

[TABLE]

Hence

[TABLE]

Furthermore, (3.1), (3.2), (3.6), (4.4) and (4.9) yield

[TABLE]

Combining (4.10) and (4.11), we obtain

[TABLE]

But from (4.3), we have

[TABLE]

Hence

[TABLE]

by (2.12) and therefore we have $\alpha=0$.

By (3.1), we have $\nu_{P_{0}}(k^{(i)}_{j})\geq 0$ and $\nu_{P_{0}}(w_{u})\geq 0$ for all $i,j,u$. According to (4.9), we get $\nu_{P_{0}}(\alpha_{2})\geq 0.$ Suppose that $\alpha_{1}\neq 0$. Taking into account that $z_{0}=z_{n_{0}}=w_{0}\neq z_{h(r)}$ for $r\geq 0$, we obtain from (4.9) that $\nu_{P_{0}}(\alpha_{1})<0$. We have a contradiction. Hence $\alpha_{1}=0$ and $\alpha_{2}=0$. From Lemma C, we conclude that $f^{(i)}_{j}=0$ for all $i,j$. Hence the system (4.7) is linearly independent over $\mathbb{F}_{b}$.

Thus (4.3) is true and $({\bf x}_{n})_{n\geq 0}$ satisfies the condition (2.6). By Theorem E, $({\bf x}_{n})_{n\geq 0}$ is the $d-$admissible uniformly distributed digital $(t,s)$-sequence in base $b$. Applying Theorem B, we get the assertion of Theorem 2.

**4.3 Generalized Halton-type sequence. Proof of Theorem 3.

Lemma 3.** *The sequence $({\bf x}_{n})_{n\geq 0}$ is uniformly distributed in $[0,1)^{s}$ .

*Proof. By Lemma A, in order to prove Lemma 3, it suffices to show that $m-T(m)\to\infty$ for $m\to\infty$. Let $R_{k}=\max_{1\leq i\leq s}n_{i,k}$, $k=1,2,...$ . We define $j_{i,k}$ from the following condition $n_{i,j_{i,k}}\geq R_{k}>n_{i,j_{i,k}-1}$. Let $\tilde{R}_{k}=\sum_{i=1}^{s}n_{i,j_{i,k}}$.

We consider the definition of $(t,m,s)$ net. Suppose that for all

[TABLE]

$1\leq i\leq s,\;d_{1}+\cdots+d_{s}=R_{k}$, we have

[TABLE]

$j\in[1,d_{i}],\;i\in[1,s]$.

By Definition 2, we get that $({\bf x}_{n})_{n\geq 0}$ is a $(T,s)$-sequence in base $b$ with $m-R(k)\geq T(m)$ for $m\geq\tilde{R}_{k}+(3g+3)e_{0}$. Bearing in mind that $R(k)\to\infty$ for $k\to\infty$, we obtain the assertion of Lemma 4.

Taking into account that $d_{i}\leq R_{k}\leq n_{i,j_{i,k}}$ for $1\leq i\leq s$, we get that in order to prove (4.12), it suffices to verify that

[TABLE]

for all $u^{(i)}_{j}\in\mathbb{F}_{b}$, with $j\in[1,n_{i,j_{i,k}}],\;i\in[1,s]$.

Let $\mathcal{M}=(m_{0}e_{0}+2g-1)P_{\infty}$ with $m_{0}=[m/e_{0}]-2g-1$.

By Lemma 1, we obtain that there exist sets $H_{1}$ and $H_{2}$ such that $H_{1}\cup H_{2}=\{0,1,...,m-1\}$, $H_{1}\cap H_{2}=\emptyset$, $(w^{(0)}_{r})_{r\in H_{1}}$ is the $\mathbb{F}_{b}$ linear basis of $\mathcal{L}(\mathcal{M})$ and $\#H_{2}=m-m_{0}e_{0}-g=:g_{1}$, with $g_{1}-e_{0}(2g+1)-g\in[0,e_{0})$. Let $n=\sum_{r=0}^{m-1}a_{r}(n)b^{r}$ and let

[TABLE]

So

[TABLE]

We fix $\tilde{n}\in\ddot{U}$. Let

[TABLE]

It is easy to see that statement (4.13) follows from the next assertion

[TABLE]

Taking into account that $y^{(i)}_{n,j}=y^{(i)}_{\dot{n},j}+y^{(i)}_{\ddot{n},j}$, we get

[TABLE]

where $\dot{u}^{(i)}_{j}=\dot{u}^{(i)}_{j}-y^{(i)}_{\tilde{n},j}$. Let

[TABLE]

We consider the map $\breve{\psi}\;:\;\mathcal{L}(\mathcal{M})\to\mathbb{F}_{b}^{\tilde{R}_{k}}$ defined by

[TABLE]

Note that $\breve{\psi}$ is a linear transformation between vector spaces over $\mathbb{F}_{b}$. It is clear that in order to prove (4.14), it suffices to verify that $\breve{\psi}$ is surjective. To prove this, it is enough to show that

[TABLE]

Using (3.9), (3.11) and (3.14), we get that $w^{(i)}_{l}\equiv 0\;({\rm mod}\;\mathcal{P}_{i,j_{i,k}})$ for

$l\geq n_{i,j_{i,k}}$. By (3.9), (3.11), (3.14), and (3.17), we derive that $y^{(i)}_{n,j}=0$ for all $j\in[1,n_{i,j_{i,k}}]$ if and only if $\dot{f}=f_{n}\equiv\;0\;({\rm mod}\;\mathcal{P}_{i,j_{i,k}})$ for $i\in[1,s]$.

From the definition of $\breve{\psi}$ it is clear that

[TABLE]

Using Riemann-Roch’s theorem, we obtain that $\dim(\mathcal{M})=m_{0}e_{0}+g=m-g_{1}$, where ${\rm deg}(\mathcal{M})=m_{0}e_{0}+2g-1$ and

[TABLE]

Hence $\dim({\rm ker}(\psi))=m-\tilde{R}_{k}-g_{1}\geq(3g+3)e_{0}-g_{1}\geq(3g+3)e_{0}-(2g+2)e_{0}-g\geq 1$, $\dim(\mathcal{M})=m-g_{1}$ and (4.15) follows. So $\breve{\psi}$ is indeed surjective. Therefore (4.14) and Lemma 3 are proved.   

Lemma 4. *The sequence $({\bf x}_{n})_{n\geq 0}$ satisfies condition (2.6).

*Proof. Let

[TABLE]

for $i\in[1,s]$, $n_{0,j_{0,m}}=([m/e_{0}]+1)e_{0}$ $j_{0,m}=[m/e_{0}]+1$.

Bearing in mind that $y^{(0)}_{n,j}=\bar{a}_{j-1}(n)$, $(j=1,2,...)$, we get from (3.18) - (3.19), that in order to obtain (2.6), it suffices to prove that

[TABLE]

for all $u^{(i)}_{j}\in\mathbb{F}_{b}$. Let $\mathcal{M}=(([M_{1}/e_{0}]+1)e_{0}+2g-1)P_{\infty}$. By (3.8), ${\rm deg}(P_{\infty})=1$. Hence ${\rm deg}(\mathcal{M})=([M_{1}/e_{0}]+1)e_{0}+2g-1$. Using Riemann-Roch’s theorem, we obtain that

[TABLE]

By Lemma 1, we get that an $\mathbb{F}_{b}$ linear basis of $\mathcal{L}(\mathcal{M})$ can be chosen from the set $\{w^{(0)}_{0},...,w^{(0)}_{M-1}\}$ with $M=([M_{1}/e_{0}]+3g+1)e_{0}=n_{0,[M_{1}/e_{0}]+3g+1}$.

Let $n=\sum_{r=0}^{M-1}a_{r}(n)b^{r}$ and let $f_{n}=\sum_{r=0}^{M-1}\bar{a}_{r}(n)w^{(0)}_{r}$. We get that for all $\dot{f}\in\mathcal{L}(\mathcal{M})$ there exists $n\in[0,b^{M})$ such that $\dot{f}=f_{n}$.

From (3.17), we have

[TABLE]

Let

[TABLE]

Consider the map $\dot{\psi}\;:\;\mathcal{L}(\mathcal{M})\to\mathbb{F}_{b}^{M_{1}}$ defined by

[TABLE]

We see that in order to obtain (4.17), it suffices to verify that $\dot{\psi}$ is surjective.

To prove this, it suffices to show that

[TABLE]

Using (3.9), (3.11) and (3.14), we get that $w^{(i)}_{k}\equiv 0\;({\rm mod}\;\mathcal{P}_{i,j_{i,m}})$ for $k>n_{i,m}$. From (4.19), (3.9), (3.11) and (3.14), we derive that $y^{(i)}_{n,j}=0$ for all $j\in[1,n_{i,j_{i,m}}]$ if and only if $f_{n}\equiv 0\;({\rm mod}\;\mathcal{P}_{i,j_{i,m}})$ for $i\in[0,s]$.

From the definition of $\dot{\psi}$ it is clear that

[TABLE]

Using (4.16), (4.18), (3.8) and Riemann-Roch’s theorem, we obtain that

[TABLE]

and $\dim({\rm ker}(\psi))=g_{1}+g$. By (4.18), $\dim(\mathcal{M})=M_{1}+g_{1}+g$. Hence \dim\big{(}\mathcal{L}(\mathcal{M})/{\rm ker}(\dot{\psi})\big{)}=M_{1}. Therefore (4.21) is true. So $\psi$ is indeed surjective and (4.17) follows. Therefore Lemma 4 is proved.   

Lemma 5. The sequence $({\bf x}_{n})_{n\geq 0}$ is weakly admissible.

Proof. Suppose that $x^{(i)}_{n}=x^{(i)}_{k}$ for some $i,n,k$. From (4.20) and (3.18)-(3.19), we get that $y^{(i)}_{n,j}=y^{(i)}_{k,j}$ for $j\geq 1$.

Using (4.19), we have

[TABLE]

Hence $f_{n}=f_{k}$. Taking into account that $(w^{(0)}_{r})_{r\geq 0}$ is an $\mathbb{F}_{b}$ linear basis of $O_{F}$, we obtain from (3.16), that $n=k$. By Definition 7, Lemma 5 is proved.

Applying Theorem B, we get the assertion of Theorem 3.   

4.4 Niederreiter-Xing sequence. Proof of Theorem 4.

Similarly to the proof of Lemma 5, we get that $({\bf x}_{n})_{n\geq 0}$ is weakly admissible. By Lemma 2, $({\bf x}_{n})_{n\geq 0}$ is the digital uniformly distributed sequence.

According to (2.2), (2.3), (2.6) and Theorem B, in order to prove Theorem 4, it is enough to verify that

[TABLE]

for all $u^{(i)}_{j}\in\mathbb{F}_{b}$, where $M=s\tau_{m}+m+2g+2$.

Bearing in mind that by Lemma 2 $(x^{(0)}_{n},{\bf x}_{n})_{n\geq 0}$ is a $(g,s+1)$ sequence, we obtain

[TABLE]

for all $u^{(i)}_{j}\in\mathbb{F}_{b}$.

Therefore, in order to prove (4.22), it suffices to verify that

[TABLE]

Now we will prove (4.23) :

From (3.21) and (3.29), we have $\ddot{v}_{r}\dot{\tau}_{0}=\sum_{j\geq 1}\dot{c}^{(0)}_{j,r}t_{0}^{j-1}$ with $\nu_{P_{0}}(\ddot{v}_{r}\tau_{0})\\ =\alpha(r)$. Hence $\dot{c}^{(0)}_{j,r}=0$ for $j\leq\alpha(r)$ and $\dot{c}^{(0)}_{j,r}\neq 0$ for $j=\alpha(r)+1$.

Using (2.2), (3.33) and (3.35) we obtain $\dot{c}^{(0)}_{j,\beta(j-1)}\neq 0$ and

[TABLE]

We apply induction and consider the case $j=1$. By (3.36), we see that $\bar{a}_{\beta(0)}(n)=0$ if $y^{(0)}_{n,1}=0$. Suppose that $\bar{a}_{\beta(0)}(n)=\cdots=\bar{a}_{\beta(l-1)}(n)=0$ if $y^{(0)}_{n,1}=\cdots=y^{(0)}_{n,l}=0$ for some $l\geq 1$ .Now let $y^{(0)}_{n,1}=\cdots=y^{(0)}_{n,l}=y^{(0)}_{n,l+1}=0$. We see

[TABLE]

Bearing in mind that $\dot{c}^{(0)}_{l+1,\beta(l)}\neq 0$, we get $\bar{a}_{\beta(l)}(n)=0$.

Therefore if $y^{(0)}_{n,j}=0$ for all $1\leq j\leq m+g+1$, then $a_{\beta(j-1)}(n)=0$ for all $1\leq j\leq m+g+1$. Using Lemma 2, we get $a_{r}(n)=0$ for all $0\leq r\leq m-1$.

Hence (4.23) is true and Theorem 4 follows.   

Aknowledgment. Parts of this work were started at the Workshop ”Discrepancy Theory and Quasi-Monte Carlo methods” held at the Erwin Schrödinger Institute, September 25 - 29, 2017.