Minimal redundant digit expansions in the gaussian integers

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

C

LEMENS

H

EUBERGER

Minimal redundant digit expansions in the gaussian integers

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{2 (2002),}

p. 517-528

<http://www.numdam.org/item?id=JTNB_2002__14_2_517_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Minimal

redundant

_{digit expansions}

in

the

Gaussian

_integers

par CLEMENS HEUBERGER

RÉSUMÉ. Un résultat récent établit

_qu’il

suffit de connaître les

deux derniers chiffres

_{significatifs}

du

_{développement}

en

_{base q}

usuel d’un entier pour calculer le dernier chiffre

significatif

dans

le

_{développement}

en

_{base q}

redondant minimal. Nous montrons que l’énoncé

analogue

pour les entiers de Gauss est faux.

ABSTRACT. We consider minimal redundant

_digit

_expansions

in

canonical number _systems in the Gaussian

_integers.

In contrast to

the case of rational

_integers,

where the

_knowledge

of the two least

significant digits

in the "standard"

_expansion

suffices to calculate the least

_{significant digit}

in a minimal redundant

_expansion,

such a _property does not hold in the Gaussian numbers: We prove

that there exist

_pairs

of numbers whose non-redundant

_expansions

agree

arbitrarily

well but which have different least

significant

digits

in minimal redundant

_expansions.

1. Introduction

Let n

_{and q &#x3E;}

2 be

_positive

rational

_integers.

Redundant q-ary expan-sions n = with

arbitrary

digits

_E Z have been studied

by

several

authors,

motivated

_{by applications}

from

_cryptography

and

_coding

theory.

For

_general

_positional

number

_systems,

we refer to Knuth

_[3,

Sec-tion

_4.1.].

An overview over results on redundant _q-ary

_digit

_expansions

is

contained in

_[1].

The aim is to minimize the cost of an

_expansion

which is

given by

Recently,

we

_proved

[1]

that the

_knowledge

of the first two

_digits

_{qo, r~l} of the "standard"

expansion n

=

E" 0

?7j qj

with 0 q suffices to decide

j=

-what

_{digit -o}

should be taken in order to achieve a minimal

_expansion

with

respect

to the costs

_(1).

_{By using}

this

_information,

we could

provide

an

efRcient

_algorithm

to

_compute

a minimal

expansion,

a formula to

_compute

a

_{single digit}

without

_having

to

_compute

the

_others,

and we _{gave estimates}

for the average costs of such an

_expansion.

A natural

_question

is whether such a result is also true if we

_replace

the _q-ary

_expansion

in the rational

_integers

_by

an

_expansion

in a canonical

number

_system

in some

algebraic

number field. In this _paper, we

_give

a

negative

answer for the case of the Gaussian

_integers.

Let R be a

subring

of the

_ring

of

_integers

in an

algebraic

number field K.

For

_#

E

_R,

_{(,8, {0, ... ,}

₁₁₎

is called a canonical number

_system

if all a E R have a

_unique

_{representation}

We refer to Kovics and Peth6

_[4]

for further discussions on canonical

num-ber

_systems.

Kitai and Szab6

_[2]

characterized canonical number

_systems

in the

Gaussian

_{integers: #}

E is a base of a canonical number

_system

if

and

_only

if

_{Re #}

0 and

_Im,Ci

= f 1.

Let

₍₃

be such a base and a E

_Z[i].

A redundant

_expansion

_of

a in base

{3

is a

_tuple

(ro, ... , ri)

with _rj E Z for

_{0 j}

I such that

A Minimal redundant

_expansions

_of

a in base

{3

is a redundant

_expansion

with minimum costs

The main result of this note is the

_following

theorem.

Theorem 1.

_Let,8

be a base

_of

a canonical number

_{s ystem}

in the Gaussian

integers.

For all L &#x3E; 0 there is a

pair

of

numbers _a,

a’

with the

following

properties:

1. Let a = and a’ =

¿j=o

aj,8J

be the

_{unique representations}

according

to

₍₂₎

with

_l,

l’ &#x3E; L. Then _aj =

aj

for 0 j

L.

2. For all

_pairs

_{(ro, ... ,}

_rs)

and

_(ro,

... ,

of

minirrzal redundant

expan-sions

_of

a and

a’,

_{respectively,}

we have

ro =1=

ro.

This

_implies

that there are numbers where the

knowledge

of the first

L + 1

_digits

in the "standard"

_expansion

₍₂₎

cannot be used to derive the first

_digit

of a minimal

_expansion.

We call a

_pair

_a, a’ as described in

According

to the result of Katai and Szab6 we have to

_consider#

= -n+I

for n &#x3E;

_1;

without loss of

_generality

we _may assume

_~3

= -n + i. We first discuss

_general

properties

of minimal

_expansions

in Section 2. We show that

not all

_integers

can occur in a minimal

_expansion.

This

_implies

that there are

usually

two choices for the least

_{significant digit}

of the

_expansion.

We

demonstrate how to _prove rules which can avoid

_branching

in some cases.

Some of these rules are based on the fact that some

_digits

cannot occur,

other rules have to be

_{proved by checking}

minimal

_expansions

of a certain

set of numbers. Then we construct a critical

_pair

for n &#x3E; 3 in Section 3.

The

_special

cases n = 1 and n = 2 are

_investigated

in Sections 4 and

_5,

respectively.

The construction of critical

_pairs

is done as follows.

First,

we

collect

_enough

rules in order to derive minimal

_expansions

of some numbers

recursively.

In a second

_step,

we take all

possible

least

significant digits

for one

_component

of the

_pretended

critical

_pair,

use the known

_expansions

from the

_previous

_step

in order to calculate the minimum costs for all alternatives and see that

_only

one of the

possible

least

significant digits

leads to a minimal

_expansion.

_By

_doing

the same for the other

_component

of the critical

_pair

we _prove that the least

significant

digits

in minimal

redundant

_expansions

differ.

For an A C Z we use the notation

A* :=

ak E A

and _ak = 0 for almost all

A:}.

We

_identify

finite _sequences

_{(ao, ... ,}

_{al )}

with the

_{corresponding}

infinite

se-quences

(ao, ... , al,

0, ... ).

The indices of all sequences start with 0 unless

otherwise stated.

For

_{given ~3}

and for a E

_Z[I]

and a E Z* we write a _’::::.,8 a if a =

Similarly,

we write

We define

opt,6 (a)

:=

{r

E Z* : r is a minimal redundant

_expansion

of a in base

{3}

and extend this notation to a E Z*

_by

_opt~(a)

:=

The concatenation of finite _sequences is denoted

_by

’

This notation is extended to concatenations of a finite _sequence a E

Z~

with a set of infinite sequences R C Z*: a &#x26; R :_

_{a

&#x26; b : b E

_RI.

_Finally,

the

_repetition

of a _sequence is defined

_by

_{(ao, ... ,}

=

(ao, ... , al ) &#x26;

(ao, ... , al )

&#x26; ... &#x26;

_(ao,

... , where the block

(ao, ... , al )

is

repeated

k

2.

_Properties

of Minimal

_Expansions

Let us first state the

_following

observation:

Not all

_integers

can occur as

_digits

in a minimal

expansion,

as is shown

in the

_following

lemma.

Then we have

Proof.

We consider first the case n &#x3E; 3. Assume that

_(7b)

is not true.

Then there is some r E

_optp(a)

and

_{some j}

such that

_lrjl

&#x3E;

n2/2

+ n + 1. is also a

- - -

-redundant

_expansion

of a for Q =

sign(rj ) .

We have

If

_lrj

&#x3E;

_M,

we conclude that

otherwise,

we

_get

This is a contradiction to the

_assumption

that r is a minimal

expansion.

Therefore,

(7b)

is

_proved

3. Because

_n2/2

_{+ n + 1 M for n &#x3E; 3,}

this

_yields

_(7a)

also.

The

_proof

of

_(7b)

for n = 1 and _n = 2 is similar: We

repeat

the above

instead of relation

_(5).

Now,

we _prove

(7a)

for n = 1. Assume that fl D* =

0

and

let r E

_By

_assumption,

there is

_{some j}

such that = 2. Let

a

:= sign(r~).

If j

l-3,

we _may

replace r

by’=

(ro, ... ,

CT, _r~+4, ... ,

ri),

since

(-2, 0,1,1)

’:::t.,8 0 and

c(r’) -

c(r)

0. We

emphasize

that r and r’ are of same

length.

Therefore we

_repeat

this _process

finitely

many times in order to obtain an r :=

(ro,..., ri)

E

_opt~(a)

such that

We may

replace

(ri-2

7ri - 1,

ri)

by

any element of

optp(r,-2, r’-l, rl).

It

can

_easily

be checked that n

_{D* ~}

0

for all choices

Finally,

we note that for n =

2,

(7a)

_can be

proved similarly using

relation

(5).

a

We note that Lemma 2 and Lemma 3 can be used to calculate one

(or all)

minimal

_expansion

in

_exponential

time. Assume that we want to

_compute

opt~(a)

for some a = a+bi. Since _{a -}

(a + nb) (mod ~3),

the _set of

_possible

least

_significant

_digits

is contained in R := n

_{(-Un, Un~.}

This

yields

An

_{implementation}

of these ideas in Mathematica can be obtained from

http://vwv.opt.math.tu-graz.ac.at/-cheub/publications/minimal

redundantgauss/.

The

_following

lemma shows how to _prove some rules of the

_following

_type:

If the standard

_expansion

₍₂₎

starts with

_digits

_{(ao, ... ,}

then there is

an

_{optimal expansion}

which starts with the

_digit

_ro. Lemma 4. Let n &#x3E;

_1,

_,(3

= _-n

+ i,

_{a E}

Dl+l,

_a

’=={3 a and _{ro E} Z with

Irol

M. Then the

_following

conditions are

_equivalent:

1. For adl t E Z* we have &#x26;

t)

_{fl ro} &#x26;

_{Z* 7~}

0.

2. For adl a’ that

_{satisfy a’ -}

a

(mod

and

for

which there is an

s E with a’ ’:::::!.,8 s we have

Note that s in condition 2 and a are of same

_{length 1}

+ 1.

Proof.

Assume condition 1. Since a’ - a

(mod

~3l+1 )

there is an

_expansion

a’ a &#x26; t for some t E Z*. Therefore there exists some s E Z* such that

Conversely,

assume condition 2. Let t E Z* and define

a"

_’:::1.,8 a &#x26; t.

Choose some s E D* n &#x26;

_t)

=

opt (a").

This intersection is

non-empty

by

(7a).

We

define a’

:= and note that

a’ -

a" m a

(mod ~3l+1).

By

assumption,

there is some l’ &#x3E; I and some r =

(rl, ... ,

rl’)

_E

Z"

(we

do not

enforce

_{rl’ =1=}

₀₎

such that

_(ro)

&#x26; r E

By

construction of r, we

_get

Since s E

_opt,

_(a"),

this _proves that

_(ro)

&#x26; u E

_{opt,8( a")}

= &#x26;

t).

D

The

_following

lemma shows that condition 2 of Lemma 4 can be checked

efficiently:

Lemma 5. Let

_{n &#x3E;_}

_1,

_,8

= _{-n +}

_i,

a, a’ E

Dl+l,

a _{’:::!.,8 a,}

a’

a’ such

that 0:’ - a

(mod

,C~l+1 .

Then there is some _E Z

i

with

- ~ I

-such

that a’

= _{a +}

Proof. Let y

E

_{Z lil}

such that a’ - a = We obtain

I

3. Critical Pair for n &#x3E; 3

The

_following

lemma calculates the minimal

_expansion

of some

_special

numbers which will occur in the construction of a critical

_pair.

Lemma 6. Let n &#x3E;

_3,

~3

= -n

_{+ i,}

and

Proof.

For R E

Z*,

Lemma 2 and

_(7b)

_imply

By

Lemma 2 and

_(7b),

an

_{optimal expansion}

of x _may start with x or

x - M.

Since Ix - At)

+ 1 &#x3E;

_Ixl

for n &#x3E;

_5,

we

proved

(l0a)

in this case. For n E

_{{3, 4},}

relation

_(l0a)

can be checked

_{directly. Similarly,}

we can

verify

(lOd).

-

-The

_proofs

of

_{(lOc), (l0e),}

and

_(lOt)

for k = 1 _are similar.

An inductive

_argument

_completes

the

_proof

of the lemma.

We are now able to construct a critical

pair:

Then

Proof.

We first consider the case 1 = 2k with k &#x3E; 2. As in

(8),

relations

(7b)

and

_(lla)

_imply

2 -

_{M, ~ -}

_2n)

&#x26;

_{(x) ~l~-l~ ~ .}

We can now use the

_expansions

calculated in

_(lOb)

and

_(lOt)

to obtain the

_following

candidates for minimal

expansions

of

_{(x - 2)}

&#x26;

_(x)~2~‘~.

Since

the first alternative has to be taken in order to minimize the costs. We

note that we

_{proved equality}

in

(lOb),

which

_yields

₍₁₂₎

for 1 = 2k &#x3E; 4.

The other cases can be

proved similarly.

D

4. Critical Pair for n = 1

In Table

_1,

we collect some choices

(a,

ro)

E

Dl+l

x D for which the conditions of Lemma 4 are fulfilled.

The

_following

lemma will be needed for the construction of a critical

pair

for n = 1:

Lemma 8. Let

_{3

- -1 +

_i,

R _E

Z*,

_{q &#x3E;}

_0,

0 _r

_5,

and s =

5q

₊ r.

TABLE 1. Some valid choices for n =

_1,

_a, and _ro in Lemma 4

Proof.

First,

we _prove that

, , 1-1

From Table 1 we derive

Using

(13)

and

_noting

that and that

Applying

this

_{relation q}

times

_completes

the

_proof.

0 We are now able to construct a critical

pair

(which

_proves Theorem 1 for

n = 1).

Proof.

From

_(7b)

we see that

_{optg (a)}

C

₍₁₎

&#x26; Z* U

_(-1)

&#x26; Z*. We note

that a can also be

represented by

((- 1, 0,

2,1, 0, 0)

&#x26;

(1, 1,

O)(5q+l»).

We use Table 1 and Lemma 8 to calculate

Similarly,

we obtain

Since

by

(15)

an

optimal

_expansion

of

((0, 1,

_0,

_0,

0)

&#x26;

(1, 1,

0) (5q+ 1) )

has

cost

_{3 + 6q}

and an

_{optimal expansion}

of

~(0, 2,1, 0, 0)

&#x26;

(1,1, O)~59+1)~

has

cost 2 +

_6q,

we

_get

(14)

for a.

Analogously,

we note that

where the

_optimal

cost is

_6q

+ 3. On the other

_hand,

a’ can be

represented

by

( -1, 0, 2,1, 0, 0)

&#x26;

_{( 1,1,}

_0)(5q+l)

&#x26;

_{(1, 1, 1),}

which

_yields

with

_optimal

cost

_6q

+ 4. This

_completes

the

_proof.

D In the case n =

_1,

Table 1 shows that in _{most cases} it is sufficient _to know

a few more

digits

in the "standard"

expansion

to derive the correct

_digit

in an

_optimal

_expansion.

Therefore,

an

_algorithm

to

_compute

an

_optimal

expansion

could

_precompute

some more entries for Table 1 and branch

_only

in those cases where no information is known.

We note that the relation

_{(-1 -f- i)4 = -4}

_strongly

relates the standard

expansion

in base -1 + i to the

_expansion

in base -4 in Z.

_However,

this observation cannot be used for minimal

_expansions

because this correspon-dence does not

_respect

the costs

_(4).

5. Critical Pair for n = 2

We will

_repeatedly

use the relation

(5, -l, -3, -1)

0 and the rules

TABLE 2. Some valid choices for n =

_2,

a, and ro in Lemma 4

Proof.

As in the

_proof

of Lemma

_8,

_{repeated application}

of rules in Table 2

yields

for R E Z*.

_Iterating

this result and

_noting

that

_{( 1, 3,1 )}

E

_{3,1 )}

proves the lemma. 0

The

_proof

of the

_following

_proposition

_completes

the

_proof

of Theorem 1 for n = 2:

Proposition

11. Let n =

_2,

~3

= -2

~- i, u &#x3E;

0. Then

Proof.

Let a

( 3, 4, 4,1 )

&#x26;

(1,3, 1)(u).

We write u : = 2s + r with r E

~ 0,1 ~ .

According

to

_(7b),

we have to consider first

_{digits - ?, - 2, 3, 8.}

All

possible

expansions

are

_given

in Table

_3,

which has been

_{computed using}

Table

_2,

Lemma 10 and an inductive

_argument

_(for

_expansions

_starting

with -7).

0

References

[1] C. HEUBERGER, H. PRODINGER, On minimal _expansions in redundant number systems:

Al-gorithms and _{quantitative analysis. Computing} 66 _(2001), 377-393.

[2] I. _KÁTAI, J. SZABÓ, Canonical number _{systems for complex integers.} Acta Sci. Math. _(Szeged) 37 _(1975), 255-260.

[3] D. E. KNUTH, Seminumerical algorithms, third ed. The Art of Computer Programming,

vol. _2, Addison-Wesley, 1998.

[4] B. _KOVÁCS, A. PETHÖ, Number _systems in integral domains, especially in orders _{of algebraic} number _fields. Acta Sci. Math. _(Szeged) 55 _(1991), 287-299.

Clemens HEUBERGER Institut f3r Mathematik

Technische Universitat Graz

Steyrergasse 30 A-8010 Graz Austria

E-mail : demons. _{heubergerttugraz.} at