The distribution of powers of integers in algebraic
number fields
par WERNER GEORG NOWAK et JOHANNES SCHOISSENGEIER
RÉSUMÉ. Pour tout corps de nombres K
(non
totalementréel) ,
se pose la question de déterminer le nombre de puissances
p-ièmes
d’entiers
algébriques 03B3
de K, vérifiant03BC(03C4(03B3p))
~ X, ceci pour toutplongement
03C4 de K dans le corps des nombrescomplexes. Ici,
X est un
paramètre
réelgrand,
p est un entier fixé ~ 2 et03BC(z) = max(|Re(z)| , |Im(z)|) (z,
nombrecomplexe).
Ce nombre est évaluéasymptotiquement
sous la formecp,KXn/p+Rp,K(X),
avec des es-timations
précises
sur le resteRp,K(X).
La démonstration utilise destechniques
issues de la théorie desréseaux,
dont enparticulier
la
généralisation multidimensionnelle,
donnée par W. Schmidt, du théorème de K.F. Roth surl’approximation
des nombresalgébri-
ques par les nombres rationnels.
ABSTRACT. For an
arbitrary (not totally real)
number field Kof
degree ~
3, we ask how manyperfect
powers03B3p
ofalgebraic integers 03B3
in K exist, such that03BC(03C4(03B3p))
X for eachembedding
03C4 of K into the
complex
field.(X
alarge
real parameter, p ~ 2 afixed
integer,
and03BC(z)
=max(|Re(z)|, |Im(z)|)
for anycomplex z.)
This quantity is evaluated
asymptotically
in the formcp,KXn/p
+Rp,K(X),
withsharp
estimates for the remainderRp,K(X).
Theargument uses
techniques
from lattice pointtheory along
withW. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation of
algebraic
numbersby
rationals.1. Introduction and statement of results
In the context of
computer
calculations on Catalan’sproblem
inOpfer
andRipken [18]
raisedquestions
about the distributionof p-th
powers of Gaussianintegers (p > 2, p
E Zfixed).
Inparticular, they
asked for theirnumber
.Mp(X)
in a square{z
E (C :Ec(z) X ~,
whereManuscrit regu le 7 octobre 2002.
X a
large
realparameter.
WhileOpfer
andRipken
gaveonly
the crudebound
Mp(X)
GX2/p,
H. Müller and the first named author[14]
under-took a
thorough analysis
of theproblem. They
reduced it to the task toevaluate the number
Ap((X)
of latticepoints (of
the standard latticeZ~)
inthe
linearly
dilated copy of theplanar
domainFIG. 1 : The domain
D2
FIG. 2 : The domainD3
In
fact,
where is a certain error term. The area of
Dp
isreadily computed (supported,
e.g.,by
Mathematica[22])
asfor p >
2,
resp.,area(D2)
=4log (1
+ = 3.52549 .... Here2F1
denotesthe
hypergeometric function
Using deep
tools from thetheory
of latticepoints’,
inparticular
those dueto
Huxley [3], [4],
it wasproved
in[14]
thatand
Furthermore,
as was discussed in detail in
[17].
Kuba
[7] - [11] generalized
thequestion
to squares ofinteger
elements inhypercomplex
numbersystems
likequarternions,
octaves, etc.In this paper we consider the -
apparently
natural -analogue
of theproblem
in analgebraic
number fieldK,
nottotally real,
ofdegree [K : Q]
= n > 3.(The
case of anarbitrary imaginary quadratic
field K can bedealt with in the same way as that of the Gaussian
field,
as far as the 0-estimate
(1.4)
isconcerned.) Again
for a fixedexponent p >
2 andlarge
realX,
letMp,K(X)
denote the number of z E K with thefollowing properties:
(i)
There exists somealgebraic integer q
E Kwith yP
= z,(ii)
For eachembedding
T of K into (C,Our
objective
will be to establish anasymptotic
formula for.Mp,K(X).
Theorem. For any
fixed exponent p > 2,
and any nottotally
real numberfield
Kof degree [K : Q]
= n >3,
where WK is the number
of
rootsof unity
inK,
r the numberof
real em-beddings
K - R, s= I (n - r),
anddisc(K)
the discriminantof
K. Theerror term can be estimated as
follows:
If
K is"totally i. e.,
r =0,
1 For an enlightening presentation of the whole topic the reader is recommended to the mono-
graphs of Kratzel [5] and [6].
where
throughout
E > 0 isarbitrary
butfcxed.
The constantsimplied
in the«-symbol
maydepend
on p, K and E.Remarks. Our method of
proof
usestechniques
from thetheory
of latticepoints
in domains with boundaries of nonzerocurvature2, along
with toolsfrom
Diophantine approximation.
It works best for n = 3 and for n =4,
K.1 +,E
totally complex.
The bounds achieved in these two cases readX p +
resp.,. / j, -Ll +,E
The first one
significantly improves
upon the estimate 0which had been established in
[12].
The result for r =0, n
> 6 is somewhat crude(and, by
the way,independent
of ourDiophantine approximation techniques). Using
a moregeometric argument
andplugging
inHuxley’s
1 (n-
100 ) method,
this mostlikely
can be refined toXp n- 73 +E
2.
Auxiliary
results and otherpreliminaries
First of
all,
we reduce our task to a latticepoint problem.
To thisend,
we have to take care of
multiple
solutions of theequation ’"’(P
= z.Since,
asis well-known
(cf.,
e.g., Narkiewicz[16],
p.100),
the set of all roots ofunity
in an
algebraic
number field K forms acyclic
group(of
order WK,say),
itis
simple
to see that K containsjust gcd(p, wx) p-th
roots ofunity,
for any fixedp >
2.Therefore,
if we defineAp,K (X )
as the number ofalgebraic integers
in E K with theproperty
that X for eachembedding
T : K -~ C, it
readily
follows thatLet ~1, ... , ar be the real
embeddings
and ar+1, cry-pi,..., ar+s, ar+s thecomplex embeddings
of K into C.Further,
let r =(c~i,..., an)
be a fixed(ordered) integral
basis of K. For u =(~c 1, ... , un )
EJRn,
we define linear formsn
and
Obviously,
is the number of latticepoints (of
the standard lattice in thelinearly
dilated copyX1/PBp
of thebody
2Cf. again the books of Kratzel (5~, [6].
In our
argument,
we shall need the lineartransformations 3 -
°and
Let P
= 2 1 1 ,
and define an(n
xn)-matrix P r,8
as follows:Along
the main
diagonal,
there are first r1’s,
then s blocksP,
and all other entries are 0. Then it issimple
to see thatwhere
M, C,
are definedby (2.2), (2.3), respectively.
SinceIdet Pi
=1,
itis clear that
For any matrix A E we shall write A* =
’A-’
for thecontragra-
dient matrix. Of course,
where r 8 is
quite
similar toP r,s, only
with Preplaced by
P* =1 1 (i -i
*From this it is easy to
verify
that the first r rows of C* are real and that its(r
+2k)-th
row is thecomplex conjugate
of the(r
+ 2k -l)-th
row, k = I1, ... ,
- I I - I B , s.Moreover,
if we define linear forms - -’Y1 , ILj (w), j
=1,...,
r + s,it
readily
follows from(2.5)
that j...,r-~ s,
Moreover,
since the linear transformation w - C*w isnon-singular,
IIC*wIl2
attains apositive
minimum = 1.By homogeneity,
for.. _ irf ~ f / ~ -,"’"
3Vectors are always meant as columns vectors although we write them - for convenience of
print - as n-tuples in one line.
A salient
point
in ourargument
will be to estimate from below absolute values of the linear formsLj (m),
m E For this purpose, we needfirst some information about the linear
independence (over
therationals)
of their coefficients.
Lemma 1. For
nonnegative integers
p, n, denoteby
the
symmetric
basicpolynomial of
index p in the n ircdeterminatesX 1,
...,Xn (in particular, so =
1 andan
algebraic
numberfield of degree I embeddings
into (C. Then the set.linearly independent
overQ.
Proof.
Choose xo, ...,Xn-1 E Q
so thatWe construct a finite sequence
polynomials , by
therecursion
The sum on the
right
hand side isalways
a rationalnumber,
andobviously
the
degree
of Pk is 1~throughout.
We shall show that for 0 l~ n,We use induction on k. For k = 0 we
simply appeal
to(*). Supposing
thathas
already
beenestablished,
wemultiply
thisidentity by
Ql(a)
and useto obtain
or, in other terms,
which, by
our recursionformula, yields just (**) .Putting k
= n in(**) gives
= 0. As Ql
(a)
hasdegree n
overQ,
it follows that pn = 0.By
ourrecursive
construction,
all pj vanishidentically, j = 0, ... ,
n. Let I be theminimal index such that
xj 54
0 in(*). Applying (**)
withinfer that
xlsnn 11~ (Q2(a), ... , an (a))
=an(a)
=0,
thus XI =0,
hence the assertion of Lemma 1. D
Lemma 2. Let T =
(a1’..., an)
be a basisof
the numberfield
K as avector space over
Q.
Let the matrix A bedefined
as A = =the
embeddings of
K into C. Thenfor
each rowof A*,
the n elements are
linearly independent
overQ.
Proof. W.l.o.g.
we may choose notation so that the assertion will be es-tablished for the
first
row of A*. We start with thespecial
case thatthe basis has the
special
formFo
=(1,
a,a2,
... ,an-1).
ThenAo
=Qa))ly k-1n
is a Vandermondematrix,
and a little reflectionshows4
-J, -
that the first row of
A* (which
is the first column ofAü 1)
isproportional
to the vector
By
Lemma1,
this isactually linearly independent
overQ.
If F isarbitrary,
and K =
Q(o:),
there exists anonsingular (n
xn)-matrix
R =with rational
entries,
so that A =AoR,
hence also A* =AoR*.
Let uswrite A* - R" = Assume
that xl, ... , zn are rational numbers with
Then
4,n fact, it is plain to see that, for j > 2, the j-th row of Ao is orthogonal to this vector.
Since we have
already
shown the linearindependence
of(~31,1, . ~ ~ , ,C3i,n),
each of the last inner sums must vanish. As
det(R*) 0 0,
thisimplies
thatzi # ... # zn # 0.
0Lemma 3. Let
L(m)
= cimi + " ’ + Cn7?2n be a linearform
whosecoeffi-
cients are
algebraic
numbers andlinearly independent
overQ.
Letfurther
~y > 1,
E > 0 and c > 0 befixed,
and let Y and M belarge
realparameters.
(i) If
L is a realform, it follows
that(ii) If
L is notproportional
to a realform,
then(iii) Furthermore,
in that latter case,Proof. (i)
Wesplit
up the sum inquestion
asBy
a celebrated result of W. Schmidt[20]
and[21],
p.152, IL(m)1
»M-~n-1~-E~,
under the conditions stated. Form, m~
Em 0 ~m~,
Ilmil., lim’ll. M,
it thus follows thatfor short.
(Here
we mimick an idea which may befound,
e.g., in the book ofKuipers
and Niederreiter[13],
p.123.)
Hence the real numbers withm E 0
~~m~~~ M,
have distances > z from each other(apart
fromthe fact that
IL(m)1 =
which does not affect the «-estimates whichfollow)
and also from 0 =I L (0, .. - , 0) ~. Therefore,
thus
(Note
that81
= 0 if11Y z.) Furthermore,
which
gives just
the assertion(i). Slightly
moregeneral (and
useful forwhat
follows),
wereadily
derive thefollowing
conclusion: If q is a fixed rationalnumber, (w)
denotes the distance of the real w from the nearestinteger, and cli, ... , ¿n-1
are realalgebraic
numbers such that1, ci,
... ,¿n-1
are
linearly independent
overQ,
then(with Y,
M asbefore),
(ii) Dealing
with thecomplex
case, weput v = n - 2,
and may as-sume
w.l.o.g.
that Cv+2 = 1 and R.For j = 1, ... , v ~- 1,
weset aj =
Recj, bj =
>and5 a = (al, ... , av),
b -(61,..., ~),
m =(m 1, ... , mv ) .
We notice thatFor m E
Z’,
we thus choose the nearestinteger
to -b ~then
mv+2 (m)
the nearestinteger
to-mv+1 (m)
m.(To
bequite precise,
the nearestinteger
to x E R is meant to be[x
+1]. Throughout, -
denotes the standard inner
product.) Accordingly,
we defineThe sum to be estimated is therefore
Now consider the group G :=
{m
E b - E and let R denoteits
rank,
0 R v.According
to Bourbaki[2], chap. VII, § 4, No. 3,
p.
18,
Theorem1,
there exist a basis(el,
... ,e,)
of Z’ andpositive integers
gl, ... , 9R, so that
(91
eel, ... ,gR eR)
is a basis ofG, i.e.,
G = git, Z
+ ... +Furthermore,
each m E Z’ has a(unique) representation
50nly
for the rest of this proof, vectors have less than n components. We symbolize this by writing m instead of m, and so on.where h =
(hl, ... , hR)
E~R,
q =(Q1,...,QR)
E~R, 0 h~
forj - 1, ... , R,
and t =(tR+1, ... , tv)
EObviously,
m E G iff allhj and tj
vanish.By
Cramer’srule,
Mimplies
thatalso 1191100
« Mand
Iltlloo
« M. As a furthersimple
consequence of(2.12),
the numbers1,
eR.+1 ~b~bv+1, ... , b/bv+1
arelinearly independent
overQ.
SinceR
is
rational,
we infer from(2.9) (with n -
1replaced by
, , , , , - 1- , ..,
v-R, the tj taking
over the role of the mj in(2.9)), (2.10) (and
that the
portion
of theright
hand side of(2.11)
whichcorresponds
tom ~
Gis
since
gl,... ,9R «1.
It remains to deal with the m E G. Here(2.12) simplifies
to ] with qunique. Hence,
in this case,Therefore
(motivated by
a look at(2.10)),
we infer thatI D I
We claim that the numbers := :=
a - ©b) R,
are
linearly independent
overQ.
Writeti
=:(e~i,... ~ ej~R)
for1 j
R.Suppose
thatkoqo
+ ... +kR’TJR
= 0 for certainintegers ko,
... ,k R.
This isjust
the real part of theequality
whose
imaginary part
is trivial. The last sum isagain
a rational num-ber. Since
1,
cl, ... , cv, cv+1 arelinearly independent
overQ,
it follows thatfor I =
1, ... ,
v, hence1~1
= ... =kR
=0,
and alsol~o
= 0.-
Therefore, by (2.9) (with R
instead of n -1), (2.10),
and the fact thatI Rezl I
the relevantpart
of theright
hand side sum in(2.11)
isCombining
the bounds(2.13)
and(2.15),
wecomplete
theproof
of clause(ii).
(iii)
For a furtherparameter
1 GM1
GM,
consider thebody
It is easy to
see6 that vol(W(M, Ml)) «MfMn-2,
hence alsoTherefore,
Finally,
we letM1
range over adyadic
sequence, 1 «M1
«M,
and sumup to
complete
theproof
of Lemma 3. 0Lemma 4. There exist
positive
constantsCl, C2, depending only
on p andon the
integral
basis rof
the nurraberfield K,
so thatfor
0 wC1
the
following
holds true: Forarbitrary points
u, v E IfBn such that u EBp, v ~ (1
+w)Bp,
itfollows
thatlIu - V112
>C2w.
Proof. By
definition ofBp,
there are twopossible
cases: EitherIAj(v)1 ]
>1 +w for
some j
E{I,
... ,r},
then6E.g.,
by a change of the coordinate system, using a basis of R’ which contains (al, ... , av+1, 1), (bl, ... , b.,+ 1, 0), with L as in (2. 10).hence and
by Cauchy’s inequality
... ’" ,.. i > . ....
By
Lemma D in17,
therefore the Euclidean, for
appropriate
Lemma 5. For the
region Dp
andarbitrary
real numbersTi, T2,
notboth
0,
- -Proof.
This isjust (a special
caseof)
Lemma 1 in[12].
03. Proof of the Theorem
Like in
[12],
weemploy
a method of W. Mfller[15],
ch.3,
to evaluatethe number of lattice
points Ap,K (X )
in thebody
For any fixedpositive
real N thereexists7
a continuous function61 :
R n -[0, oo[
withthe
following properties:
(i)
Thesupport of 61
is contained in the n-dimensional unit 1.r
M.
(iii)
The Fourier transform satisfieswhere u - w denotes the standard inner
product.
For a smallparameter
w >
0,
weput 8~,(u)
=u).
Then thesupport
ofðw
is containedin the
ballllul12 c.,
and it follows thatFor an explicite construction of such a 61, one can start from the normalized indicator function of a smaller ball and apply repeated convolution with itself. See also [12] for further
details and references.
Denote
by Is
the indicator function of any set ,S’ CR’ -
We shall use convolution to smoothen indicator functions:We
put
withC2
as in Lemma4,
and claim:I. . Then there exist u* E
Bp,
Lemma
4,
I 1 G
, therefore
(a)
is true.Similarly,
. There exist u* E R’ v* E with
.
Again by
Lemma4,
thus 1
which is
equivalent
to(b).
Since 0 G *6~
1throughout, (a)
and(b) imply
thatTherefore,
By
Poisson’s formula in R’(see
Bochner[1]),
we thus infer thatSince I and
we obtain from
(3.1)
and(3.2)
with
n
:=Z" B{(0,..., 0 . Writing
Y instead ofY±
for short(thus
Y ~X1/P),
we have to estimateTo this
integral
weapply
the linear transformation u - v = Mu defined in(2.2).
Thuswhere
"’
Using throughout
the notation introduced in section2,
inparticular (2.6),
we can write this as
-r
As an immediate
by-result,
weobtain, by
anappeal
to(2.4),
By
Lemma5, (3.5) implies (for m ~ (0, ... , 0))
in view of
(2.7). Combining
this with(3.3),
we arrive at°
We divide the range of summation of this sum into subsets
where J E
f 1, ... ,
r +~},
and M islarge. By (2.8), always
We have to
distinguish
two cases.Case 1: r =
0, i. e.,
K is"totally complex".
Letw.l.o.g.
J =1,
then weconclude that
by (3.9)
and anapplication
of Lemma3, (ii)
and(iii).
As a consequence, thecorresponding portion
of the sum in(3.8)
can be estimatedby
We fix N
sufhciently large
and let M runthrough
the powers of 2. It follows thatRecalling
that K xX’IP,
andbalancing
the remainderterms,
weget
w =1 112
and thus
For
larger
n, it is favorable to choose a different(actually
somewhatcruder)
approach.
For any U E{I,..., ,5}
andparameters
1 CCTl, ... , Tu
CCM,
we consider the
body
It is
plain
to see that the number ofpoints
of the lattice M*7Gn (with
M*as in
(2.2) through (2.6))
inKu (Ti,
... ,Tu)
satisfiesTherefore,
We let the
Tk
range overdyadic
sequences «M,
and sum also over U =1, ... , s
to arrive atWith N fixed
sufficiently large,
we letagain
M range over the powers of 2 to obtainTaking
we thusget
Note that, because of (3.9), this sum would be empty for U = 0, if M is sufficiently large.
Case 2: r > 1.
Dealing again
with the sum in(3.8),
we consider first those domainsS(M, J)
with J Ef r
+1, ... ,
r +s},
say, J = r + s.Similarly
to(3.10),
by
anappeal
to(3.9)
and Lemma3, part (i). Secondly,
we deal withS(M, J)
where J Ef 1, ... , r 1,
say, J = 1. Thecorresponding argument
now reads
again
in view of(3.9)
and Lemma3, (ii)
and(iii). Therefore,
thepart
ofthe sum in
(3.8)
whichcorresponds
Again,
with fixed lVsufhciently large,
we let M rangethrough
the powers of2,
and sum up to infer from(3.8)
thatRecalling
that Y = andbalancing
the remainderterms,
we choose1 1
W = X-p n-2 and thus obtain
Combining
the estimates(3.11), (3.12),
and(3.15),
andrecalling (3.6)
and(2.1),
wecomplete
theproof
of our Theorem. DReferences
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Werner Georg NOWAK
Institute of Mathematics
Department of Integrative Biology
BOKU - University of Natural Resources and Applied Life Sciences Peter Jordan-StraBe 82
A-1190 Wien, Austria
E-mail : nowakQmai1 .boku . ac . at Johannes SCHOISSENGEIER
Institut fur Mathematik der Universitat Wien
Nordbergstralie 15
A-1090 Wien, Austria
E-mail : hannes.schoissengeier@univie.ac.at