\documentclass[12pt]{article}
\input{defs2}
\def\noi{\vskip10pt\noindent}
%\usepackage{pifont}


\title{Basic Digit Sets and Their Fractions Sets}
 \author{Dr. Darren Wick \\
 Ashland University}
 \date{October 9, 2007}
 
 \begin{document}
 \maketitle
\noi
\cl {\bf The Usual Suspects}
\noi
1) Base $b=10$ with digit set $D=\{0,1,\ldots ,9\} $
$$(284.93)_{10}=2\cdot10^2 +8\cdot 10^1 +4\cdot 10^0 +9\cdot 10^{-1} +3\cdot 10^{-2}$$
$$1 =(.\ol {9})_{10}$$
\noi
\noi
\noi
2) Base $b=2$ with digit set $D=\{ 0,1\}$
$$35 = 1\cdot 2^5 +1\cdot 2^1+1\cdot 2^0 =(100011)_2$$
$$ 1 = {{1\over 2}\over {1-{1\over 2}}}=\sum_{k=1}^{\infty}
1 ({1\over 2})^k =(.\ol 1)_2$$
\noi
\noi
\noi
3) Base $b=60$ with digit set $D=\{ 0,1,\ldots ,59\}$
\noi
\cl {\bf Preliminaries}
\noi
Let $b\in \mathbb {Z}$  and $D\subset \mathbb{Z}$ 
\noi
 A representation of a number $z$ in the {\it base} $b$ with {\it digit set }$D$ is
$$z=\sum_{k=-\infty}^l a_k b^k $$
for some $l\in \mathbb{Z}$ and all $a_k \in D$
\noi
$$z=(a_l ,a_{l-1} , \cdots ,a_1,a_0 {\bf . } a_{-1},a_{-2}, \ldots)_{b}$$
\noi
$$a_l b^l +\cdots +a_1 b +a_0 \ = {the \ integer \ part \ of \ z}$$
\noi
$$a_{-1} b^{-1} + a_{-2} b^{-2} + \cdots  ={the \ fractional \
part \ of \ z}$$ 
\noi
Denote the base and digit set pair as $(b,D)$
\noi

\cl {\bf Questions}
\noi
When can all real numbers be represented?
\noi
When can all integers be represented uniquely (as integers)?
%(as an integer!)?
\noi
%\cl {Answers}
%1) often
%\noi
%2) about as often
%\noi
\cl {\bf Examples}
\noi
1) In the usual base $b= 10, \ D=\{ 0,1,\ldots ,9\}$
\noi
$$\ds -{1\over 2} = -(.5)_{10}
\ {\rm and \ we \ need \ a \ "sign \ bit"}$$
\noi
$$\pm 0 \ {\rm and} \ 1=.\overline{9} \ {\rm have \ redundant \ representations}$$
\noi
2) With $b=10$ and $D=\{ -2, -1, 0, \ldots ,8\}$
%$$19=(2,-1)$$
$$-{1\over 2} = (-1.5)_b \ {\rm requires \ no \ sign \ bit}$$
\noi
3) Balanced Ternary: $$b=3 \ \ \ \ \  D=\{ -1,0,1 \}$$
\noi
$$( .{\ol {1}})_b = {1\over 3}+{1\over 9} +\cdots ={{1\over 3}\over {1-{1\over 3}}}={1\over 2}$$
\noi
$$( .{\ol {-1}})_b = -{1\over 3}-{1\over 9} +\cdots =-{1\over 2}$$
\noi
4) Negabinary: Base $b=-2$ and
$D=\{0,1\}$
\noi
$$ -1=(11)_b $$
\noi


{\bf Goal}: (Matula) The characterization and computation of those
integral-valued base and digit set pairs that provide complete and unique
finite radix representation of the integers without the need of a sign bit.
\noi
{\bf Definition}:
Let $b\in \mathbb{Z} $, $|b|\geq 2$ and $0\in D$. Then $(b,D)$ is  {\it basic} if every integer $z$ has a unique representation of the form $$z=\sum_{k=0}^l a_k b^k$$ with
each $a_k \in D$.
\noi
Example: The usual base 10 is not basic, since negative integers
have no such representation.
\noi
 Example: $(10,\{ -2,-1,0, \ldots
,8\})$ is not basic since
$80$ has two integral representations,
$$(8,0)_b \ {\rm and} \ (1,-2,0)_b$$
\noi
Example: Negabinary $(-2,\{ 0,1 \} )$ and balanced 
 ternary $(3,\{
-1,0,1\} ) $ are basic.
\noi
\cl {\bf
Question} What are the basic sets? If $b$ is positive, then we must
have some negative digits in $D$.
\noi
{\bf Theorem} (Matula): $(b,D)$ is basic iff

   i) $D$ is a complete residue system modulo $b$ (in
particular, $|D|=|b|$)

   ii) For each $n\in \fatn$, the set $$D^n=\{ \sum_{i=0}^{n-1} a_i b^i : a_i\in D\}$$ contains no non-zero multiples of $b^n -1$.
\noi
\cl {\bf Examples}

\noindent
1) The usual base 10 with $n=1$ fails part ii
\noi
2) $(10,\{-2,-1 ,\ldots ,8\})$ fails part i
\noi
3) $(10,\{-1 ,\ldots ,8\})$ is OK
\noi
4) Negabinary and Balanced Ternary are OK
\noi
\cl {\bf Question}  Do the rest of the real numbers 
have a representation in a basic set? Is it unique?
\noi
{\bf Theorem }(Matula): If $(b,D)$ is basic then every real number
has a representation
%2) Balanced ternary: $(3,\{-1,0,1 \} )$ is basic

\cl {Example}
\noi
Negabinary $(-2,\{0,1\})$ is basic
$$-{1\over 3}={1\over 4}{4\over 3}-{1\over 2}{4\over 3}=-{1\over 2} \bigg({1\over {1-{1\over 4}}}\bigg)+{1\over 4} \bigg({1\over {1-{1\over 4}}}\bigg)$$
$$=-{1\over 2} (1+{1\over 4}+{1\over {16}}+\cdots \cdots ) +{1\over 4} (1+{1\over 4}+{1\over {16}}+ \cdots )$$$$=(-{1\over 2})^1+(-{1\over 2})^2+(-{1\over 2})^3+(-{1\over 2})^4+\cdots =(.\ol 1)_b$$
\noi
%2) Balanced ternary: $(3,\{-1,0,1 \} )$ is basic
\noi

\cl {\bf Fraction Sets}
\noi
For a given base $(b,D)$, denote by $F=F(b,D)$ the set of all  numbers that have a representation with zero integer part.
\noi
For the usual base 10, F is the interval [0,1] that
\begin{enumerate}
\item has unit length
 \item
 is connected
\item
 covers the real line by 
unit translations
\item
has endpoints $\pm 0$ and $1=.\ol 9$ with redundant representations
\end{enumerate}
\noi
\noindent
{\bf Example}: 
For $b=5$ and $D=\{ -1,0,1,2,3\}$, the fraction set appears to be
$[-{{1}\over {4}},{3\over {4}}]$
\noi
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Real/base5(-1,0,1,2,3).pdf}

%\includegraphics[height=3in,width=5in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/base51.pdf}

\noindent
Note
 that
the endpoints acquire redundant representations upon translation of the
interval one unit in either direction since
$$-{1\over 4}= (.\ol{-1})_b =-1+{3\over 4}=-1+(.\ol{3})_b=(-1.\ol 3 )_b$$

$${3\over 4}= (.\ol{3})_b =1+-{1\over 4}=1+(.\ol{-1})_b=(1.\ol {-1})_b 
$$

\noi\noindent
{\bf Example}: 
For $b=-5$ and $D=\{-1, 0,1,2,3\})$ the fraction set is appears to be $F=[-{2\over 3} , {1\over 3}]$ 

\noi

%\includegraphics[height=3in,width=5in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/base-51.pdf}
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Real/base-5(-1,0,1,2,3).pdf}

\noi\noindent
{\bf Example}: For $b=5$ and $D=\{ -3, -1,0,1,3\} $, it appears that the fraction set is disconnected. Note how the unit translates must cover the real line.

\noi
%\includegraphics[height=3in,width=5in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/base52.pdf}

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Real/base5(-3,-1,0,1,3).pdf}

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Real/base5(-3,-1,0,1,3)trans.pdf}

\noi\noindent
{\bf Example}: For $b=3$ and $D=\{ 0,1,-25\} $, it appears that the fraction set is disconnected.

\noi
%\includegraphics[height=3in,width=5in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/base32.pdf}


\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Real/base3(0,1,-25).pdf}



\noi
\noindent
{\bf Lemma 1}: Suppose $(b,D)$ is basic. Let $a=min(D)$ and $A=max(D)$. Then
\begin{enumerate}
\item If $b>0$, then $$min(F)= \frac{a}{b-1} \ \ \ \  \ max(F)= \frac{A}{b -1}$$
\item If $b<0$, then $$min(F)= \frac{Ab+a}{b^2-1} \ \ \ \ \ max(F)= \frac {ab+A}{b^2 -1}$$
\end{enumerate}
\noi
{{\bf Theorem }: If $(b,D)$ is a basic set with all consecutive digits in $D$, then $F$ is an interval of length 1.  In particular, 
$$F=[min(F),max(F)]$$
\noi
{\bf Corollary}: If $(b,D)$ is a basic set with all consecutive digits in $D$, then the endpoints of $F$ have redundant representations.
\noi
\noi
{\bf Theorem}: If $(b,D)$ is basic and $D$ does not consist of consecutive digits, then $F$ is disconnected.
\noi
QUESTIONS: 
Does F always have measure one?
What are the redundant representations in general?
Can we characterize the connected subsets of F?


%\cl {Conjectures}
%\noi
%1) For basic digit sets of sequential integers of magnitude $\leq |b|$, the
%fraction set is a closed interval of unit length.
%\noi
%2) For basic  digit sets containing non-sequential integers and with all
%digits of magnitude $\leq |b|$, the fraction set is disconnected with
%"measure" one. There is also some kind of fractal boundary since these
%intervals cover the real line via translation.
%\noi
%Graphs of some Real Fraction Sets and their Translates
%\picture 0in by 2.5in (base-5.1 scaled 0.45)
%\picture 0in by 2.5in (base3.2 scaled 0.45)
\cl {\bf Complex Bases and their Fraction Sets}
\noi
The Usual Suspect: Two copies of standard base 10
\noi
\noi
A base $(b,D)$ for the complex numbers is {\underbar {valid}} 
if every Gaussian integer $z=n + mi$ can be written uniquely
$$z=\sum_{k=0}^l a_k b^k $$
for some $l\in \fatn$ and all $a_k \in D$.
\noi
\noi
Theorem (Gilbert): If $(b,D)$ is valid, then
\noi {} i) every complex number has a representation
\noi {} ii) $D$ is a complete residue system for the Gaussian integers modulo
$b$
\noi {} iii) $|D|=|b|=$, where for $b=n+mi$, $|b|=n^2+m^2$
\noi
Question: What are equivalence classes in the complex numbers?
\noi
For $b\in \fatc$, one example of a complete residue system modulo $b$
is $D=\{0,1,2\ldots ,|b|-1 \}$
\noi
Let $b=-2+i$. Then $D=\{0,1,2,3,4 \}$ is complete. $4+i \equiv 1$ modulo $b$ since $4+i -1=3+i$ and
$${{3+i}\over {-2+i}}={{(3+i)(-2-i)}\over {(-2+i)(-2-i)}}={{-5-5i}\over
5}=-1-1i$$
is a Gaussian integer.
\noi
Let $b=2+i$. Then $D=\{ 0,1,-1,i,-i \}$ is  complete. For example, $6+7i\equiv -i$ modulo $b$ since $6+7i-(-i) =6+8i$ and
$${{6+8i}\over {2+i}}={{(6+8i)(2-i)}\over {(2+i)(2-i)}}={{20+10i}\over
5}=4+2i$$
is a Gaussian integer.
\noi
Theorem: (Davio, Deschamps, and Gossart) There exists a valid base for all Gaussian
integers of modulus greater than 1 except for bases $2$ and $1\pm i$.
\noi
\noi
Theorem: (Katai, Szabo) Let $b$ be a Gaussian integer of the form $-n \pm i$ 
for $n\geq 1$.
Let
$D=\{0,1,2,\ldots ,n^2 \}$. Then $(b,D)$ is valid.
\noi
Plots of fraction sets of various  bases with $b$ a Gaussian integer and
$D$ the digit set $$D=\{ 0,1,\ldots ,|b|-1 \}$$
$b=-1+i$
\vskip0pt
%\picture 0in by 2in (base-1+i scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Gaussian/-1+i.pdf}


\noi
\noi
$b=-2-i$
\vskip0pt
%\picture 0in by 2in (base-2-i scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Gaussian/-2-i.pdf}
\noi
$b=-2+i$
\vskip0pt
%\picture 0in by 2in (base-2+i scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Gaussian/-2+i.pdf}

\noi
%$b=-3-i$
%\noi
%%\picture 0in by 2in (base-3-i scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Gaussian/-3-i.pdf}


\noi
$b=-3+i$
\vskip0pt
%\picture 0in by 2in (base-3+i scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Gaussian/-3+i.pdf}
\noi
$b=-4+i$
\vskip0pt
%\picture 0in by 2in (base-4+i scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Gaussian/-4+i.pdf}

\noi
$b=-5+i$
\noi
%\picture 0in by 2in (base-5+i scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/Gaussian/-5+i.pdf}

\noi
\noi
\noi

\noi
Theorem (Gilbert): If $(b,D)$ is a valid base for $\fatc$,
then the fraction set is closed with unit area and tiles the plane by
translations using the Gaussian integers. Points on the boundary lie on the
boundary of multiple translates and thus have redundant representations.
\noi
\noi
\noi
$$(-2+i,\{ 0,1,2,3,4\} )$$
\vskip0pt
%\picture 0in by 2in (base-2+i.tile scaled 0.5)

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/-2+i/translate.pdf}

\noi
%Example: Fraction set for $(-2+i ,\{0,1,2,3,4\} )$
\noi
$(-2+i,\{ 0,1,2,3,4\} )$ all fractions with digits 0,1
\vskip0pt
%\picture 0in by 2in (base-2+i.2dig scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/-2+i/2dig0,1.pdf}

\noi
 $(-2+i,\{ 0,1,2,3,4\} )$ all 2 digit fractions
\vskip0pt
\noi
%\picture 0in by 2in (base-2+i.2digall scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/-2+i/2digall.pdf}

\noi
 $(-2+i,\{ 0,1,2,3,4 \} )$ all fractions with digits 1,2,3
\vskip0pt
%\picture 0in by 2in (base-2+i.3dig scaled 0.5)

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/-2+i/3dig1,2,3.pdf}

\noi
 $(-2+i,\{ 0,1,2,3,4 \} )$ all 3 digit fractions
\vskip0pt
%\picture 0in by 2in (base-2+i.3digall scaled 0.5)

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/-2+i/3digall.pdf}

%*****************FIX THIS*****************
 $(-2+i,\{ 0,1,2,3, 4\} )$ all fractions
\vskip0pt
%\picture 0in by 2in (base-2+i.all scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/-2+i/alldigits.pdf}

\noi
\cl {Example (Knuth)}
\noi
The base $(2i,\{0,1,2,3\})$ is not valid since $i=(10.2)$ is not an
integer. However, every complex number can be represented in a unique way in
this base! All bases of the form $(\sqrt {-n},\{ 0,1,2,\ldots ,n-1\})$ have
this property of completely unique representations. However, in the case
$b=\sqrt 2 i$, $i$ itself has an infinite representation.
\noi
\cl {Powers of $2i$}
$$
\matrix{(2i)^{-5}&(2i)^{-4}&(2i)^{-3}&(2i)^{-2}&(2i)^{-1} & (2i)^0\cr
-{1\over {32}}i & {1\over {16}} & {1\over 8}i & -{1\over 4}& -{1\over 2}i &1\cr
}$$
$$
\matrix{(2i)^{1}&(2i)^{2}&(2i)^{3}&(2i)^{4}&(2i)^{5} & (2i)^6\cr
2i & -4 & -8i & 16 & 32i & -64\cr
}$$
\noi
$$(11210.31)_{2i} =
$$$$1\cdot (2i)^4+1\cdot (2i)^3+2\cdot (2i)^2+ 1\cdot
(2i)+3\cdot (2i)^{-1}+1\cdot (2i)^{-2}$$$$=-7.5i+7.75$$
\noi
$(2i,\{ 0,1,2,3\} )$ all fractions with digits 0,1
\vskip0pt
%\picture 0in by 2in (base2i.1.2dig scaled 0.5)

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2iReal/2dig0,1.pdf}

\noi
 $(2i,\{ 0,1,2,3\} )$ all 2 digit fractions
\vskip0pt
%\picture 0in by 2in (base2i.1.2digall scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2iReal/2digall.pdf}


\noi
$(2i,\{ 0,1,2,3\} )$ all fractions with digits 1,2,3
\vskip0pt
%\picture 0in by 2in (base2i.1.3dig scaled 0.5)

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2iReal/3dig1,2,3.pdf}

\noi
 $(2i,\{ 0,1,2,3\} )$ all 3 digit fractions
\vskip0pt
%\picture 0in by 2in (base2i.1.3digall scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2iReal/3digall.pdf}

\noi
 $(2i,\{ 0,1,2,3\} )$ all fractions
\vskip0pt
%\picture 0in by 2in (base2i.1.all scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2iReal/alldigits.pdf}


\noi
 $(2i,\{ 0,1,-1,i,-i\} )$ all 2 digit fractions
\vskip0pt
%\picture 0in by 2in (base2i.2.2digall scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2iComplex/2digall.pdf}

\noi
 $(2i,\{ 0,1,-1,i,-i\} )$ all 3 digit fractions
\vskip0pt
%\picture 0in by 2in (base2i.2.3digall scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2iComplex/3digall.pdf}

\noi
 $(2i,\{ 0,1,-1,i,-i\} )$ all fractions
\noi
%\picture 0in by 2in (base2i.2.all scaled 0.5)

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2iComplex/alldigits.pdf}

\noi
\noi

 $(2+i,\{ 0,1,-1,i,-i\} )$ all 2 digit fractions
\vskip0pt
%\picture 0in by 2in (base2+i.2digall scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2+i/2digall.pdf}

 $(2+i,\{ 0,1,-1,i,-i\} )$ all 3 digit fractions
\vskip0pt
%\picture 0in by 2in (base2+i.3digall scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2+i/3digall.pdf}

\noi
 $(2+i,\{ 0,1,-1,i,-i\} )$ all 4 digit fractions
\vskip0pt
%\picture 0in by 2in (base2+i.4digall scaled 0.5)
\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2+i/4digall.pdf}


\noi
$(2+i,\{ 0,1,-1,i,-i\} )$ 
\vskip0pt
%\picture 0in by 2in (base2+i.tile scaled 0.5)

\includegraphics[height=2in,width=2in]{/home/dwick/DATA/Professional/Research/Talks/Base_Talk_seminar07/Graphics/PartialDigits/2+i/translate.pdf}

%\cl {Conjectures}
%\noi
%1) For basic digit sets of sequential integers of magnitude $\leq |b|$, the
%fraction set is a closed interval of unit length.
%\noi
%2) For basic  digit sets containing non-sequential integers and with all
%digits of magnitude $\leq |b|$, the fraction set is disconnected with
%"measure" one. There is also some kind of fractal boundary since these
%intervals cover the real line via translation.
%\noi
%Graphs of some Real Fraction Sets and their Translates
\noi
\cl {\bf {Partial Classification of Basic Sets}}
\noi
1) There are no basic sets for $b=2$
\noi
2) For $b=-2$, the only basic sets are $$\{0,1\} \ \ \ \  \{ -1,0\}$$
\noi
3) There are infinitely many basic sets for any $|b|\geq 3$
\noi
   i) For $b\geq 3$, $(0,1,2, \ldots , b-2,(-b^k +b -1)\}$ 
is basic for all  $k\in \fatn$.
\noi
  
For $b=3$, $k=1,2,3$ we get the basic sets  $$\{ 0,1,-1 \} \ \ \ \  
\{ 0,1,-7 \} \ \ \ \ \{ 0,1,-25 \}$$
\noi
   ii) For $b\leq -3$, $(0,1,2, \ldots , |b|-2,(-|b|^{2k} +|b| -1)\}$ 
is basic for all  $k\in \fatn$.
\noi
   For $b=-3$, $k=1,2,3$ we get the basic sets  $$\{ 0,1,-7 \} \ 
\ \ \ \{ 0,1,-79 \} \ \ \ \ \{ 0,1,-727 \}$$
\noi
4) If we restrict the digit sets so that $|d|\leq |b|$ for all $d\in D$, 
then we have the following:
\noi
   i) For $b\geq 3$, $(b,D)$ is basic iff $$\{ -1,0,1 \} \subset D$$
There are $2^{b-3}$ such basic sets.
\noi
   ii) For $b\leq -3$, $D$ must contain exactly one of the subsets 
$$\{-1,0,1\} \ \ \ \{-(b+1),0,1\} \ \ \  \{-1,0,b+1\}$$ There are
$\ds 3\cdot 2^{|b|-3}$ such basic sets.
\noi
\cl {\bf Examples}
   i) The restricted basic digit sets for $b=5$ are
$$\{ -3,-2,-1,0,1 \} \ \ \ \{ -2,-1,0,1,2 \} $$ $$\{ -1,0,1,2,3 \} \ \ \
\{ -3,-1,0,1,3\} $$
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   ii) The restricted basic digit sets for $b=-4$ are
$$\{ -3,-2,-1,0 \} \ \ \ \{ -2,-1,0,1 \} \ \ \ \{-1,0,1,2 \} $$ $$\{ 0,1,2,3\} \ \ \
\{ -3,-1,0,2 \} \ \ \ \{ -2,0,1,3 \} $$
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