On a bounded remainder set for a digital Kronecker sequence

TL;DR

This paper characterizes when certain rectangular sets are bounded remainder sets for digital Kronecker sequences, linking this property to the finiteness of the non-zero digits in the base-b expansion of the set boundaries.

Contribution

It provides a necessary and sufficient condition for rectangular sets to be bounded remainder sets in digital Kronecker sequences, connecting this to the digit expansion of the set boundaries.

Findings

Rectangular sets are bounded remainder sets iff their boundary digits are finitely many.

The characterization applies to digital Kronecker sequences in any base b ≥ 2.

The result extends understanding of distribution properties of digital 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 digital Kronecker-sequence in base $b \geq 2$, ${\bf \gamma} =(\gamma_1,...,\gamma_s)$, $\gamma_i \in [0, 1)$ 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}

Equations

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

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

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

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

$$L=\sum _{k=w}^{\infty }{u}_k{z}^{-k}\mathrm{with}{u}_k\in {\mathbb{Z}}_b,w\in \mathbb{Z}\mathrm{where}{u}_w\ne 0.$$

$$\nu (L):=-w,L\ne 0,\nu (0):=-\infty .$$

$$\{L\}=\sum _{k=\max (1,w)}^{\infty }{u}_k{z}^{-k}.$$

$$n=\sum _{r=0}^{\infty }{a}_r(n)b^r$$

$$n(z)=\sum _{r=0}^{\infty }{\psi }_r(a_r(n))z^r\in {\mathbb{F}}_b[z]$$

$${\eta }^{(i)}(L)=\sum _{k=\max (1,w)}^{\infty }{\eta }_{i,k}(u_k)b^{-k}.$$

$${\mathbf{l}}_n=(l_n^{(1)},...,l_n^{(s)}),l_n^{(i)}={\eta }^{(i)}(n(z)L_i(z)),\mathrm{for}1\le i\le s,n\ge 0.$$

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

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

$$y_n^{(i,m)}=(y_{n,1}^{(i,m)},...,y_{n,m}^{(i,m)}),y_{n,j}^{(i,m)}=\sum _{r=0}^{m-1}{\psi }_r(a_r(n))c_{j,r}^{(i,m)}\in {\mathbb{F}}_b$$

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

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

$$[x{]}_m=\sum _{1\le j\le m}{x}_j{b}^{-j}\mathrm{with}m\ge 1.$$

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

$$E(\alpha ):=exp(2\pi i\mathrm{Tr}(\alpha )/p),\alpha \in {\mathbb{F}}_b,$$

$$\delta (\mathfrak{T})=\{\begin{array}{ll}1,& \mathrm{if}\mathfrak{T}\mathrm{is}\mathrm{true},\\ 0,& \mathrm{otherwise}.\end{array}$$

$$\frac{1}{q}\sum _{\beta \in {\mathbb{F}}_b}E(\alpha \beta )=\delta (\alpha =0),\mathrm{where}\alpha \in {\mathbb{F}}_b.$$

$$|\Delta ([0,\mathbf{\gamma }),({\mathbf{x}}_n{)}_{n=b^{m}A}^{b^{m}A+N-1})-\Delta ([0,[\mathbf{\gamma }{]}_{{\tau }_m}),({\mathbf{x}}_n{)}_{n=b^{m}A}^{b^{m}A+N-1})|\le s,\forall N\in [1,b^m].$$

$$B=[0,\mathbf{\gamma }),B_i=\prod _{1\le j\le s,j\ne i}[0,{\gamma }^{(j)})\times [0,[{\gamma }^{(i)}{]}_{{\tau }_m})\mathrm{and}{B}_0={\cup }_{i=1}^s(B\setminus B_i).$$

$$\Delta ([0,\mathbf{\gamma }),({\mathbf{x}}_n{)}_{n=b^{m}A}^{b^{m}A+N-1})$$

$$=\Delta ([0,[\mathbf{\gamma }{]}_{{\tau }_m}),({\mathbf{x}}_n{)}_{n=b^{m}A}^{b^{m}A+N-1})+\Delta (B_0,({\mathbf{x}}_n{)}_{n=b^{m}A}^{b^{m}A+N-1}).$$

$$|\Delta ([0,\mathbf{\gamma }),({\mathbf{x}}_n{)}_{n=b^{m}A}^{b^{m}A+N-1})-\Delta ([0,[\mathbf{\gamma }{]}_{{\tau }_m}),({\mathbf{x}}_n{)}_{n=b^{m}A}^{b^{m}A+N-1})|$$

$$\le \sum _{i=1}^s|\Delta (B\setminus B_i,({\mathbf{x}}_n{)}_{n=b^{m}A}^{b^{m}A+N-1})|.$$

$$x_{n+b^{m}A,j}^{(i)}=x_{k+b^{m}A,j}^{(i)}\mathrm{for}j=1,...,{\tau }_m.$$

$$y_{n+b^{m}A,j}^{(i)}=y_{k+b^{m}A,j}^{(i)}\mathrm{for}j=1,...,{\tau }_m,$$

$$y_{n+b^{m}A,j}^{(i)}=y_{n,j}^{(i)}+y_{b^{m}A,j}^{(i)},\mathrm{and}{y}_{k+b^{m}A,j}^{(i)}=y_{k,j}^{(i)}+y_{b^{m}A,j}^{(i)}\mathrm{for}j=1,...,{\tau }_m.$$

Full text

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Let $({\bf x}_{n})_{n\geq 0}$ be an $s-$dimensional digital Kronecker-sequence in base $b\geq 2$, $\boldsymbol{\gamma}=(\gamma_{1},...,\gamma_{s})$, $\gamma_{i}\in[0,1)$ 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]

Key words: bounded remainder set, digital Kronecker sequence.

2010 Mathematics Subject Classification. Primary 11K38.

1 Introduction

1.1. Discrepancy. 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 $\limsup\nolimits_{N\to\infty}N(\ln N)^{-s}\emph{D}^{*}(({\bf x}_{n})_{n=0}^{N-1})>0$. See [Bi] and [Le1] for the results on this conjecture.

An $s$-dimensional sequence $(({\bf x}_{n})_{n\geq 0})$ is of low discrepancy (abbreviated l.d.s.) if $\emph{D}^{*}(({\bf x}_{n})_{n=0}^{N-1})=O(N^{-1}(\ln N)^{s})$ for $N\rightarrow\infty$. For examples of l.d.s. see, e.g., in [BeCh], [DiPi], [Ni].

1.2. Digital Kronecker sequence.

For an arbitrary prime power $b$, let $\mathbb{F}_{b}$ be the finite field of order $b$, $\mathbb{F}_{b}^{*}=\mathbb{F}_{b}\setminus 0$, $\mathbb{Z}_{b}=\{0,1,...,b-1\}$. Let $\mathbb{F}_{b}[z]$ be the set of all polynomials over $\mathbb{F}_{b}$, and let $\mathbb{F}_{b}((z^{-1}))$ be the field of formal Laurent series. Every element $L$ of $\mathbb{F}_{b}((z^{-1}))$ has a unique expansion into a formal Laurent series

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The discrete exponential evaluation $\nu$ of $L$ is defined by

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Furthermore, we define the “fractional part“ of $L$ by

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We choose bijections $\psi_{r}:\mathbb{Z}_{b}\to\mathbb{F}_{b}$ with $\psi_{r}(0)=0$, and for $i=1,2,...,s$ and $j=1,2,...$ we choose bijections $\eta_{i,j}:\mathbb{F}_{b}\to\mathbb{Z}_{b}$. For n = 0, 1, …, let

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be the digit expansion of $n$ in base $b$, where $a_{r}(n)\in\mathbb{Z}_{b}$ for $r\geq 0$ and $a_{r}(n)=0$ for all sufficiently large $r$.

With every n = 0, 1, . . ., we associate the polynomial

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and if $L\in\mathbb{F}_{b}((z^{-1}))$ is as in (1.3), then we define

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In [Ni], Niederreiter introduced a non-Archimedean analogue of the classical Kronecker sequences. For every $s$-tuple ${\bf L}=(L_{1},...,L_{s})$ of elements of $\mathbb{F}_{b}((z^{-1}))$, we define the sequence $S({\bf L})=({\bf l}_{n})_{n\geq 0}$ by

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This sequence is sometimes called a digital Kronecker sequence (see [LaPi, p.4]). In analogy to classical Kronecker sequences, in [LaNi, Theorem 1], the following theorem has been proven

Theorem A. *A digital Kronecker sequence $S(L)$ is uniformly distributed in $[0,1)^{s}$ if and only if $1,L_{1},...,L_{s}$ are linearly independent over $\mathbb{Z}_{b}[x]$.

By $\mu_{1}$ we denote the normalized Haar-measure on $\mathbb{F}_{b}((z^{-1}))$ and by $\mu_{s}$ the $s$-fold product measure on $\mathbb{F}_{b}((z^{-1}))^{s}$. In [La1], Larcher proved the following metrical upper bound on the star discrepancy of digital Kronecker sequences $D_{N}(S(L))=O(N^{-1}(\log N)^{s}(\log\log N)^{2+\epsilon})$. For $\mu_{s}$-almost all $L\in\mathbb{F}_{b}((z^{-1}))^{s}$, $\epsilon>0$.

In [LaPi, p.4], Larcher and Pillichshammer were able to give corresponding metrical lower bounds for the discrepancy of digital Kronecker sequences $D_{N}(S(L))\geq c(b,s)N^{-1}(\log N)^{s}\log\log N$ for $\mu_{s}$-almost all $L\in\mathbb{F}_{b}((z^{-1}))^{s}$, for infinitely many $N\geq 1$ with some $c(b,s)>0$ not depending on N.

1.3. Bounded remainder set.

Definition 1. *Let ${\bf x}_{0},{\bf x}_{1},...$ be a sequence of point 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 $N$.

Let $\alpha$ be an irrational number, let I be an interval in $[0,1)$ with the 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 were studied by Lev and Grepstad [GrLe]. The case of Halton’s sequence was studied by Hellekalek [He]. For references to others investigations on bounded remainder set see [GrLe].

Let $\boldsymbol{\gamma}=(\gamma_{1},...,\gamma_{s})$, $\gamma_{i}\in(0,1)$ with $b$-adic expansion $\gamma_{i}=\gamma_{i,1}b^{-1}+\gamma_{i,2}b^{-2}\\ +...$, $i=1,...,s$. In this paper, we prove

Theorem. *Let $({\bf l}_{n})_{n\geq 0}$ be a uniformly distributed digital Kronecker sequence. The set $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is of bounded remainder with respect to $({\bf l}_{n})_{n\geq 0}$ if and only if *

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In [Le2], we proved similar results for digital sequences described in [DiPi, Sec. 8]. Note that according to Larcher’s conjecture [La2, p.215], the assertion of the Theorem is true for all digital $(t,s)$-sequences in base $b$.

2 Notations.

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. ([DiPi, Definition 4.30]) *For a given dimension $s\geq 0$, 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 $m\geq 1$ 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 $b$.

A $({\bf T},s)$-sequence in base $b$ is called a strict $({\bf T},s)$-sequence in base $b$ if for all functions ${\bf U}:\mathbb{N}_{0}\to\mathbb{N}_{0}$ with ${\bf U}(m)\leq m$ for all $m\in\mathbb{N}_{0}$ and with ${\bf U}(m)<{\bf T}(m)$ for at least one $m\in\mathbb{N}_{0}$, it is not a $({\bf U},s)$-sequence in base $b$.

Definition 4. ([DiNi, Definition 1]) 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 $n$. For $r=0,1,...$ we choose bijections $\psi_{r}:\mathbb{Z}_{b}\to\mathbb{F}_{b}$ with $\psi_{r}(0)=0$, and for $i=1,2,...,s$ and $j=1,2,...$ we choose bijections $\eta_{i,j}:\mathbb{F}_{b}\to\mathbb{Z}_{b}$. 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 $m=\infty$.

Lemma A ([LaNi, ref. 1-8]). *A digital Kronecker sequence in base $b$ can be expressed as some digital $({\bf T},s)$-sequence in base $b$.

*Lemma B ([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 $\liminf_{m\to\infty}(m-{\bf T}(m))=\infty$.

For $m>n$, we put $\sum_{j=m}^{n}c_{j}=0$ and $\prod_{j=m}^{n}c_{j}=1$. For $x=\sum_{j\geq 1}x_{j}b^{-j}$, where $x_{i}\in\mathbb{Z}_{b}=\{0,...,b-1\}$, we define the truncation

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If ${\bf x}=(x^{(1)},...,x^{(s)})\in[0,1)^{s}$, then the truncation $[{\bf x}]_{m}$ is defined coordinatewise, that is, $[{\bf x}]_{m}=([x^{(1)}]_{m},...,[x^{(s)}]_{m})$.

For $x=\sum_{j\geq 1}x_{j}b^{-j}$ and $y=\sum_{j\geq 1}y_{j}b^{-j}$ where $x_{j},y_{j}\in\mathbb{Z}_{b}$, 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 $v_{j}\in\mathbb{Z}_{b}$. For ${\bf x}=(x^{(1)},...,x^{(s)})\in[0,1)^{s}$ and ${\bf y}=(y^{(1)},...,y^{(s)})\in[0,1)^{s}$, 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 $x=\sum_{j\geq 1}x_{j}b^{-j}$, where $x_{j}\in\mathbb{Z}_{b}$, $x_{j}=0$ $(j=1,...,k)$ 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 5. A sequence $({\bf x}_{n})_{n\geq 0}$ in $[0,1)^{s}$ is weakly admissible in base $b$ if

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Let $p$ be a prime, $b=p^{\kappa}$,

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where ${\rm Tr}:\mathbb{F}_{b}\to\mathbb{F}_{p}$ denotes the usual trace of an element of $\mathbb{F}_{b}$ in $\mathbb{F}_{p}$.

Let

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By [LiNi, ref. 5.6 and ref. 5.8], we get

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3 Proof

Lemma 1. Let $({\bf x}_{n})_{n\geq 0}$ be a weakly admissible digital sequence in base $b$, $m\geq 1$, $\tau_{m}=[\log_{b}(\kappa_{m})]+m$. Then we have for all integers $A\geq 1$

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Proof. Let

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It is easy to see that $B=[{\bf 0},[\boldsymbol{\gamma}]_{\tau_{m}})\cup B_{0}$. By (1.1), we get

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Hence

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Suppose that there exist $i\in[1,s]$, $k,n\in[0,b^{m}),\;k\neq n$ and $A\geq 1$ such that $x_{n+b^{m}A},x_{k+b^{m}A}\in B\setminus B_{i}$. Therefore

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From (1.5), (2.1) and (2.2), we have

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Hence

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Therefore

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By (2.3) we have a contradiction. Thus

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Using (3.1), we get the assertion of Lemma 1.

Let $\beta_{1},...,\beta_{\kappa}$ be a $F_{p}$ basis of $\mathbb{F}_{b}$, and let ${\rm Tr}$ be a standard trace function. Let

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We use notations (1.5), (2.1) and (2.2). Let $n=\sum_{r\geq 0}a_{r}(n)b^{r}$ be the $b$-adic expansion of $n$, and let

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Therefore

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Hence

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and

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Let

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Bearing in mind that $\psi_{m-j}(a_{m-j}(n))=\omega^{-1}(a_{m-j}(\tilde{n}))$, we put

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Let

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We abbreviate $s+1$-dimensional vectors $(u^{(1)}_{n},...,u^{(s+1)}_{n})$, $(k^{(1)},...,k^{(s+1)})$ and $(r^{(1)},...,r^{(s+1)})$ by symbols $\boldsymbol{\mathsf{u}}_{n}$, $\boldsymbol{\mathsf{k}}$ and $\boldsymbol{\mathsf{r}}$, and $s$-dimensional vectors $(u^{(1)}_{n},...,\\ u^{(s)}_{n})$, $(k^{(1)},...,k^{(s)})$ by symbols ${\bf u}_{n}$ and ${\bf k}$.

By (3.2) - (3.8), we get $u^{(s+1)}_{n}=u^{(s+1)}_{n+b^{m}A}$, $A=1,2,...$,

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Let $N\in[1,b^{m}]$, $\gamma^{(s+1)}=N/b^{m}$, $k=\sum_{j=1}^{\tau_{m}}k_{j}b^{-j}>0$, with $k_{j}\in\mathbb{Z}_{b}$,

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Similarly to [Ni, Theorem 3.10] (see also [DiPi, Lemma 14.8]), we consider the following Fourier series decomposition of the discrepancy function :

Lemma 2. Let $A\geq 1$ be an integer, $N\in[1,b^{m}]$, $\gamma^{(s+1)}=N/b^{m}$, and let $({\bf x}_{n})_{n\geq 0}$ be a digital sequence in base $b$. Then

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[TABLE]

where

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$\hat{\mathds{1}}^{(i)}(0)=[\gamma^{(1)}]_{\tau_{m}}\;(1\leq i\leq s)$, $\hat{\mathds{1}}^{(s+1)}(0)=\gamma^{(s+1)}\;\;$ and

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Proof. Let $\gamma=\sum_{j=1}^{\dot{m}}\gamma_{j}b^{-j}>0$, $w=\sum_{j=1}^{\dot{m}}w_{j}b^{-j}$, with $\gamma_{j},w_{j}\in\mathbb{Z}_{b}$. It is easy to verify (see also [Ni, p. 37,38]) that

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By (2.2) and (3.6), we have that

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and

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Similarly, we derive

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Let $k\cdot u^{(i)}_{\tilde{n}}=\sum_{j=1}^{\tau_{m}}k_{j}u^{(i)}_{\tilde{n},j}$. By (2.5), we have

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Hence

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Applying (3.12) and (3.14), we derive

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Hence

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Similarly, we obtain from (3.12) and (3.13) that

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Using (3.11), we obtain

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Bearing in mind that $x^{(s+1)}_{n}=\{n/b^{m}\}$ and $\gamma^{(s+1)}=N/b^{m}$, we have

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From (3.15) and (1.1), we derive

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Hence Lemma 2 is proved.

Let

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$G_{m}^{*}=G_{m}\setminus\{{\bf 0}\}$, and let

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It is easy to see that

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Lemma 3. Let $({\bf x}_{n})_{n\geq 0}$ be a digital sequence in base $b$. Then

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Proof. By (3.12) we have $\hat{\mathds{1}}({\bf 0})=\prod_{i=1}^{s+1}[\gamma^{(i)}]_{\tau_{m}}$. Applying Lemma 2, we get

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Using (3.3), (3.6) and (3.9), we obtain

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Now from (3.4), we get (3.18). Hence Lemma 3 is proved.   

Lemma 4. *Let $({\bf x}_{n})_{n\geq 0}$ be a digital sequence in base $b$. Then *

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Proof. Using (3.6) and (3.8) and (3.11) , we have

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where

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By (3.4), (1.5) and (2.5), we obtain

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Now from (3.16), we get that $\boldsymbol{\mathsf{k}}\in D_{m}$ and Lemma 4 follows.

Let

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We consider the following conditions :

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and

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for some finite set $R_{m}$.

Bearing in mind (3.6), we get

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Lemma 5. Let $({\bf x}_{n})_{n\geq 0}$ be a weakly admissible uniformly distributed digital $({\bf T},s)$-sequence in base $b$, satisfying to (3.22) for all $m\geq m_{0}$ with some $m_{0}\geq 1$. Then (3.23) is true for $R_{m}=M_{m}$.

Proof. By (3.21) and (2.5), we obtain

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Using (3.18), we derive

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It is easy to see that if condition (3.22) is true, than ${\rm card}(M_{m})=b^{s\tau_{m}}$ and $\{\boldsymbol{\mathsf{u}}_{\widetilde{b^{m}A}}\;|\;A\in M_{m}\}=\Lambda_{m}$.

Applying (3.21), (3.25), (3.26) and (3.23) with $R_{m}=M_{m}$, we have

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Let $\dddot{n}=\ddot{n}\ominus\dot{n}$. From (1.5), we obtain $\{\dddot{n}\;|0\leq\ddot{n}<b^{m}\}=\{0,1,...,b^{m}-1\}$.

By (3.3) - (3.9), we get $\boldsymbol{\mathsf{u}}_{\dddot{n}}=\boldsymbol{\mathsf{u}}_{\ddot{n}}-\boldsymbol{\mathsf{u}}_{\dot{n}}$. Hence

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We get $\dot{\boldsymbol{\mathsf{k}}}-\ddot{\boldsymbol{\mathsf{k}}}=(0,...,0,\dot{k}^{(s+1)}-\ddot{k}^{(s+1)})$.

From (3.8), we have $u^{(s+1)}_{\tilde{n},j}=\omega^{-1}(a_{m-j+1}(n))$ and

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Taking into account (2.5), we get

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Hence $\dot{\boldsymbol{\mathsf{k}}}=\ddot{{\bf k}}$. Using Lemma 4, we obtain

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Therefore Lemma 5 is proved.   

Let $\Psi$ be a set of all bijections $\dot{\psi}:\mathbb{Z}_{b}\to\mathbb{F}_{b}$, $\psi\in\Psi$, $k\in\mathbb{F}_{b},\\ c\in\mathbb{Z}_{b}$, $A_{k,c,\psi}=E(-k\psi(c))\sum_{b=0}^{c-1}E(k\psi(b))$, $\;\langle x\rangle=\min(\{x\},1-\{x\})$, $x\in[0,1]$ and let

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By (3.12) and (3.27), we get

[TABLE]

with $\tilde{k}=-\mu\dot{k}_{v(\dot{k})}$, $\breve{k}=-\mu\ddot{k}_{v(\ddot{k})}$, $\tilde{c}=\gamma^{(s+1)}_{v(\dot{k})}$, $\breve{c}=\gamma^{(s)}_{v(\ddot{k})}$, $\psi_{1}=\psi^{-1}_{v(\dot{k})}$, $\psi_{2}=\eta^{-1}_{s,v(\ddot{k})}$,

$x_{1}=\{b^{v(\dot{k})}\gamma^{(s+1)}\}$, $x_{2}=\{b^{v(\ddot{k})}[\gamma^{(s)}]_{\tau_{m}}\}$.

Lemma 6. With the notations as above, there exist $\mathsf{a}_{1},...,\mathsf{a}_{b+7}\in\mathbb{Z}_{b}$, $\mathsf{a}_{1}^{2}+...+\mathsf{a}_{b+5}^{2}>0$, $\mathsf{a}_{b+6}=\mathsf{a}_{b+7}=0$ such that

[TABLE]

and

[TABLE]

Proof. Let

[TABLE]

Taking into account that ${\rm card}(\Psi)=b!$, we get ${\rm card}(\dot{A})\leq b!b^{2}+2<b^{b+4}$.

Let

[TABLE]

and let $z_{{\bf a}}=\mathsf{a}_{1}/b+\cdots+\mathsf{a}_{b+7}/b^{b+7}$. By (3.27), we derive

[TABLE]

Suppose that (3.29) is not true. Then for all ${\bf a}\in\ddot{A}$ there exist $k({\bf a}),c({\bf a}),\psi({\bf a})$ and $y({\bf a})$ such that

[TABLE]

Hence $|\theta_{k({\bf a}),c({\bf a}),\psi({\bf a})}+z_{{\bf a}}|<b^{-b-6}$. Suppose that $\theta_{k({\bf a}_{1}),c({\bf a}_{1}),\psi({\bf a}_{1})}=\theta_{k({\bf a}_{2}),c({\bf a}_{2}),\psi({\bf a}_{2})}$ for some ${\bf a}_{1},{\bf a}_{2}\in\ddot{A}$, ${\bf a}_{1}\neq{\bf a}_{2}$. Hence $|z_{{\bf a}_{1}}-z_{{\bf a}_{2}}|<b^{-b-5}$. Bearing in mind that $|z_{{\bf a}_{1}}-z_{{\bf a}_{2}}|\geq b^{-b-5}$ for all ${\bf a}_{1}\neq{\bf a}_{2}$, we get a contradiction. Therefore $\theta_{k({\bf a}_{1}),c({\bf a}_{1}),\psi({\bf a}_{1})}\neq\theta_{k({\bf a}_{2}),c({\bf a}_{2}),\psi({\bf a}_{2})}$ for all ${\bf a}_{1},{\bf a}_{2}\in\ddot{A}$, ${\bf a}_{1}\neq{\bf a}_{2}$. Thus ${\rm card}(\dot{A})\geq{\rm card}(\ddot{A})$. Hence

[TABLE]

We have a contradiction. Therefore (3.29) is true.

Now we consider assertion (3.30). If $c=0$, then $|B_{k,c,\psi}(x)|=|x|$ and (3.30) follows.

Now let $c\in\{1,...,b-1\}$. By (3.27), we have

[TABLE]

Using (2.5), we get

[TABLE]

[TABLE]

Taking into account that $\psi(0)=0$, we obtain

[TABLE]

Now from (3.31), we derive

[TABLE]

and (3.30) follows. Thus Lemma 6 is proved.   

Applying (3.28) - (3.30), we have

Corollary. Let $\mathsf{a}_{1},...,\mathsf{a}_{b+7}$ be integers chosen in Lemma 6 and let $\gamma^{(s+1)}_{v(\dot{k})+j}=\mathsf{a}_{j}$, $j=1,...,b+7$, with some $\dot{k}$. Then

[TABLE]

and

[TABLE]

Lemma 7. *Let $({\bf x}_{n})_{n\geq 0}$ be a digital sequence in base $b$ and let $\rho\in[2,m-2]$ be an integer. Then there exists $\boldsymbol{\mathsf{k}}\in D_{m}^{*}$ such that $k^{(1)}=...=k^{(s-1)}=0$, $k^{(s)}_{v(k^{(s)})}=1$, $1\leq v(k^{(s)})\leq\rho-1$ and $v(k^{(s+1)})\leq m-\rho+2$ *.

Proof. From (3.3)-(3.8), (3.16) and (3.20), we get that $\boldsymbol{\mathsf{k}}\in D_{m}^{*}$ if and only if

[TABLE]

We put $k^{(1)}=...=k^{(s-1)}=0,\;k^{(s)}_{j}=0$, for $j\geq\rho$ and $k^{(s+1)}_{j}=0$, for $j>m-\rho+2$. Hence (3.34) is true if and only if

[TABLE]

Therefore, in order to obtain the statement of the lemma, it is sufficient to show that there exists a nontrivial solution of the following system of linear equations

[TABLE]

In this system, we have $m+1$ unknowns $k_{1}^{(s)},...,k_{\rho-1}^{(s)}$, $k_{1}^{(s+1)},...,k_{m-\rho+2}^{(s+1)}$ and $m$ linear equations. Hence there exists a nontrivial solution of (3.36). By (3.36), we get that if $k^{(s)}=0$, then $k^{(s+1)}=0$. Hence $k^{(s)}\neq 0$ and $1\leq v(k^{(s)})\leq\rho-1$. Taking into account that if $\boldsymbol{\mathsf{k}}\in D_{m}$ then $\mu\boldsymbol{\mathsf{k}}\in D_{m}$ for all $\mu\in\mathbb{F}_{b}^{*}$. Therefore there exists $\boldsymbol{\mathsf{k}}\in D_{m}^{*}$ such that $k^{(s)}_{v(k^{(s)})}=1$ and $1\leq v(k^{(s)})\leq\rho-1$. Thus Lemma 7 is proved.   

Proposition. *Let $({\bf x}_{n})_{n\geq 0}$ be a weakly admissible uniformly distributed digital $({\bf T},s)$-sequence in base $b$, satisfying to (3.22) for all $m\geq m_{0}\geq 1$. Then $[0,\gamma_{1})\times...\times[0,\gamma_{s})$ is of bounded remainder with respect to $({\bf x}_{n})_{n\geq 0}$ if and only if (1.9) is true.

*Proof. The sufficient part of the Theorem and of the Proposition follows directly from the definition of $({\bf T},s)$ sequence and Lemma B. We will consider only the necessary part of the Theorem and of the Proposition.

Suppose that (1.9) does not true. Then

[TABLE]

Let, e.g.,

[TABLE]

Let

[TABLE]

Bearing in mind that $\{j\geq 1\;|\;\gamma_{l}^{(s)}=b-1\;\forall\;l>j\}=\emptyset$, we obtain that ${\rm card}(W)=\infty$.

Suppose that there exists $H>0$ such that $b^{2H}c_{1}>4H^{2}$, $c_{1}=\gamma_{0}^{2}b^{-4b-36}$,

[TABLE]

with $[{\bf 0},\boldsymbol{\gamma})=[0,\gamma_{1})\times\cdots\times[0,\gamma_{s})$, $\gamma_{0}=\gamma_{1}\gamma_{2}\cdots\gamma_{s-1}$.

Let $W=\{\dot{w}_{j}\;|\;\dot{w}_{i}<\dot{w}_{j}\;{\rm for}\;i<j,\;\;j=1,2,...\}$ and let

[TABLE]

We choose $m$ and $J$ from the following conditions

[TABLE]

Applying Lemma 1 and (3.38), we have

[TABLE]

By Lemma 7, we get that there exists a sequence $(\boldsymbol{\mathsf{k}}(j))_{j=1}^{J}$ such that

[TABLE]

[TABLE]

We see that the sequence $(\boldsymbol{\mathsf{k}}(j))_{j=1}^{J}$ does not depend on $\gamma^{(s+1)}$.

Using (3.37) and (3.39), we obtain $\gamma^{(s)}_{r(j)}\neq 0$. Hence

[TABLE]

$j=1,...,J$. Let $H_{1}=\{1,2,...,J\}$ if

[TABLE]

for all $1\leq j<j_{1}\leq J$, and let $H_{1}=\{j\}$ if there exist $1\leq j<j_{1}\leq J$ such that (3.44) is false. Let $\mathsf{a}_{1},...,\mathsf{a}_{b+7}$ be integers chosen in Lemma 6 and let

[TABLE]

From Lemma 5, (3.21), (3.23), (3.41) and conditions of the Proposition, we have

[TABLE]

Taking into account (3.17), (3.42) and (3.44), we get that if $\boldsymbol{\mathsf{k}}(j)\in D_{m}$ then $\mu\boldsymbol{\mathsf{k}}(j)\in D_{m}$ for $\mu\in\mathbb{F}_{b}^{*}$, and if $j_{1},j_{2}\in H_{1}$, $j_{1}\neq j_{2}$, then $\mu_{1}\boldsymbol{\mathsf{k}}(j_{1})\neq\mu_{2}\boldsymbol{\mathsf{k}}(j_{2})$ for all $\mu_{1},\mu_{2}\in\mathbb{F}_{b}^{*}$.

According to (3.11), (3.12) and (3.46), we have

[TABLE]

[TABLE]

From Corollary and (3.45), we obtain

[TABLE]

By (3.43), we can apply Corollary with $\dot{r}=r(j)-v(k^{(s)}(j))+2$. Hence

[TABLE]

Using (3.46)-(3.47) and (3.40), we obtain

[TABLE]

[TABLE]

Suppose that ${\rm card}(H_{1})=J$. From (3.40) and (3.42), we get

[TABLE]

We have a contradiction. Now let ${\rm card}(H_{1})=1$.

By (3.44), we obtain that there exist $j,j_{1}\in[1,J]$ such that $j\in H_{1}$, $j<j_{1}$ and $|v(k^{(s+1)}(j))-v(k^{(s+1)}(j_{1}))|\leq b+7$.

According to (3.39) and (3.42), we have

[TABLE]

Applying (3.38), (3.41) and (3.48), we get

[TABLE]

with $c_{1}=\gamma_{0}^{2}b^{-4b-36}$. We have a contradiction. By (3.49), the Proposition is proved.   

Completion of the proof of the Theorem. By Lemma A, $S({\bf L}^{(m)})$ is a uniformly distributed digital $(T,s)$-sequence in base $b$.

By Theorem A, we get that $1,L_{1},...,L_{s}$ are linearly independent over $\mathbb{F}_{b}[z]$. Hence $1,z^{m}L_{1},...,z^{m}L_{s}$ are linearly independent over $\mathbb{F}_{b}[z]$. Let ${\bf L}^{(m)}=(z^{m}L_{1},...,z^{m}L_{s})$, and let $S({\bf L}^{(m)})=({\bf l}^{(m)}_{n})_{n\geq 0}$ (see (1.8)) with

[TABLE]

Using Theorem A, we obtain that $S({\bf L}^{(m)})$ is a uniformly distributed sequence in $[0,1)^{s}$. Therefore, for all ${\bf w}\in\Lambda_{m}$ there exists an integer $A\geq 1$ with

[TABLE]

Thus $S({\bf L}^{(m)})$ satisfies the condition (3.22).

Bearing in mind that $1,L_{1},...,L_{s}$ are linearly independent over $\mathbb{F}_{b}[z]$, we get that $\{n(z)L_{i}\}\neq 0$ for all $n\geq 1$. Hence $\{l^{(i)}(n)\}\neq 0$ for all $n\geq 1$ $(i=1,...,s)$. Therefore the sequence $S({\bf L})$ is weakly admissible.

Applying the Proposition, we get the assertion of the Theorem.