479
Internat. J. Math. & Math. Sci.
(1978)479-496
Vol.
CHARACTERISTIC APPROXIMATION PROPERTIES OF QUADRATIC IRRATIONALS
W. B. JURKAT AND A. PEYERIMHOFF
Universitat ULM (MNH)
Abt. f. Mathematic I
7900 Ulm (Donau)
Oberer Eselsberg, Germany
(Received April 19, 1978)
ABSTRACT.
Some characteristic approximation properties of quadratic
irrationals are studied in this paper.
of the sequence
n
It is shown that the limit points
form a subset C(x), and D(x) can be generated from
C(x) in a relatively simple way.
is given using relations between
Another proof of Lekkerkerker’s theorem
dn-l’ n’ n+l which
are independent of
x and n.
KEY WORDS AND PHRASES. Quadratic Iionals, Approximation of numbers,
Badly Appro ximable Numb e
AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES.
10F05, 10F35.
480
W. B. JURKAT & A. PEYERIMOFF
O.
l’hroughout this paper
will denote a real irrational number.
x
r (x)
Given
n
[_ig,gi)
r<x)
which implies
Inxl I<i/
D(x)
x
is
lxll
ontains bounded subsequences (e.g.
by Hurwitz’s theorem), and it seems natural
n
of all its limit points which_ describes the various
qualities of approximation of
number
n e q
for infinitely many
to investigate the set
+
x
Ir(x)
nl{nxl{,
the sequence
x
x
by rationals which occur again and again
"well approximable" if
0 e D(x)
(e.g.
a Liouville number) and "badly approximable" if
[0, )
D(x)
We introduce
x=e=2.71.., or if
if
D(x)
0
1).
0 e D(x)
If
A
x
then
is
2)
hence interesting numbers in this context are the badly approximable
numbers.
Let
x
be represented by the continued fraction [b
o’ b i’
let
A /B
n
n
denote
its convergent s and let
6n 6n(X)
BnlBnX Anl
The limit points of the sequence
6
n > -2
Bn[IBnXll
(6n
C(x)
form a subset
n
constructive) and we shall show that
D(x)
for n > i). (I)
(which is in a sense
can be generated from
relatively simple way (Theorem i), so the structure of
C(x)
C(x)
in a
is basic in our
context.
A theorem of Lekkerkerker [5] lhow that for a badly approximable number
set
C(x)
is finite if and only if
x
C(x)
I)
infD(x)
which is the inverse of
and the bibliography of this paper.
2)
D(x)
D(x)
show that
is discrete if and only if
For results on
[5,
Let
Th,n
see
[i]
nil [nixll
ni
ni
0
/
n
i
choose
]]nixll
the
is a quadratic irrational, and the con-
nection between
and
x
O< e
1
for i large and
Perron’s modular
and let
ni
n
i
/
function
n
Hence
D(x).
PROPERTIES OF QUADRATIC IRRATIONALS
x
We wll also give another
(badly approximable and) a quadratic irrational.
is
proof of Lekkerkerker’s theorem using relations between
are independent of
n
and
x
[b
[b o’ b i
x
B
Pn
-
THE BASIC FORMULAS.
Writing
n
i/ Po
n > i
B
n-i
0
+ 1 ]
,bn_ 1
o’ b I
n > 0
we have for
b
n
n
+
[b
n
n
formulas
E
’n-l’ n’ n+l
which
and seem to tell the whole structure of the
n
’s (Lemma 3, Theorem 3).
I.
481
n
bn+l
the following well known
I
(2)
n+l
(-l)n
B (B x- A
n n
n
and
n+l
(3)
+1
n
bn+l
(cf.
For
p
2
n
1
b
On_ I
b
Bn+ I bn+ 1 Bn + Bn_l,
I
n > -i).
unless
3)
p
6n_l
(5)
1
b
1
n
(6)
n
It follows from (2) and (4) that
PROOF.
If
n
n > 1
+ 6n_l < 1
6n
3)
(4)
p
13; (4) is a consequence of
[7]
LEMMA i.
n+
1
Pn+l
n
1
+
i
n+l
then
+
6o
1
1
Pn- 1
n+l
+ 61
(x-Ix])
+
Pn
This and (i),
n >- i
(x-Ix]- I)
l0
(3)
W. B. JURKAT & A. PEYERIMHOFF
482
show that
n+
n-i
n+l
+Pn
I +
0n+ 1
which implies (5) (note that
for
In order to prove (6) we note that the
> 1)
n+l
(7)
n > 1
foregoing calculations also show that
I-4
Pn n+l
1-4
nn-i
(i
+
Pnn+l
2
and this leads immediately to (6).
Formulas (4) and (6) suggest the introduction of the function
l-xz +2z
(x,y;z)
4yz < 1
4xz < 1
z > 0
using this notation, we have
bn+l
(n-I
n > 0
%+i
The following properties of
(8)
0
I
will be used in later sections of this paper
(x,y;z)
(y,x;z)
(x,y;z) + (strictly)if
(x,-z;z)
(x O;z)
(
(x,i-z;z)
12z-l
x+
y+
z+
or
(io)
W_.’4.’
+ /’
2z
.i,-,..z-i,ii
2z
(9)
1
if
1-__z
z
< 1
z _< I/2
if
z > i
/i
In conclusion we mention that (5) contains Vahlen’s result (see e.g. [7]
12)
14)
PROPERTIES OF QUADRATIC IRRATIONALS
that at least one of
n-1, n’ n+l
least one of
I/
is <
11
+
5
Pn+l
[7],14) that
follows from (6), (8) and (i0).
’s
if this were not true then one of the
and (6) would imply
and Borel’s result (see
1/2
is <
n-I
n
483
would be >
i
but
2
p
i/
n
(since
Indeed,
n+l
n
is rational) and this and
(8) and (I0) imply
b
but
2.
n+l
> 1
C(x)
THE RELATION BETWEEN
In addition to
d(x)
C(x)
and
AND
D(x).
we introduce the sets
D (x)
the limit points of the sequence
n r(nx)
C (x)
the limit points of the sequence
B r(B x)
s
s
n
n
These sets contain information on the sign of the approximations of
rationals
Let
and
lnxll
D(x)
or
nx-ml,
C(x)
is known if
sign (nx- m)
n
%B
k
D (x)
s
or
C (x)
s
x
by
is known.
Then it follows from
e
+ DBk_ 1
(13)
m
by
Cramer’s
rule that
%
% + -i
k _> -I
and that
%, g
nlXBk_ 1 _i
+ (-I) k
e
Bk_ 1 llnxll
(14)
nlxBk
THEOREM i.
Let
0
-l)k
I
D(x)
Then
e
B
k
D (x)
Inxl[
if and only if
at
W. B. JURKAT & A. PEYERIMHOFF
484
%2y_
y
q
%
where
(% ) # (0 O)
0
lira
%
2B
lira B
k
1
(15)
_ir(Bk1 _1 x)
k.1 +
BY
Formula (15) and
COROLLARY.
B
and
for some sequence
Bkir(Bk’X)1
+
signy
< 0
show that
D(x)
C(x)
and
are con-
netted by
(16)
PROOF of Theorem i.
Let
ni(nix-mi)
nir(nix
a
/
Ds(X)
and select
(for all large
lq
ki
i)
such that
Bk. Inixll
Define numbers
<
follows from (17) and (14) that
lnixll
to a contradiction to
lBk.+IXll
>
Bk.+l
+
which implies
(Bk.+l
Bk.+2
n. -<
It follows from 0
+
>
(17).
ni, mi,
%i i
IBk.Xll
(17)
i
by (13) (with
%i’ i
since otherwise
lBk.Xll
nil Bk.+ixl
n
lq
o
(18)
k
i
instead of
Condition (17) implies
<
Hence, there is a constant
-<
n
i
13) and (18) that
Im IBk.+ixll Bk.+Inil Inixll
D(x)
k
by Lagrange’s Theorem ([7], 15) which lead:
([7]
<- n.2
1/2
B
It
On the other hand, it follows from
Bk.+2 )-I
21
n, m, k).
Bki+2
and
C
for all large
1
Bkl BkXll < bk+ I
C(, x)
such that
B
k
+i(II
+ o(i))
i
([7],
13)
that
bk+
0(i).
PROPERTIES OF QUADRATIC IRRATIONALS
Bk. -< ni <
C(
x)B
485
(19)
for all large i,
k .-i
From (19) and (14) we infer that
0<l
i
<
KI(
K I, K
for constants
0 -<
x)
and all large
2
-<
i
K2 (’
x)
i
By taking subsequences, the foregoing shows that sequences
n.l
/
=, k.
/
exist such that
nir(nix)
n
(20)
/
e
%Bk. + Bk.-l’
i
Bk._l r(Bk._ix)
Let
n_.,x kol
nir(nix)
satisfy (20).
2Bkir(Bkix)
+
/
REMARKS.
D (x)
s
lel
I.
< K
Bki r(B
8
k
C(x)
x)
-i’
/
’
,o (’ ) # (0, 0)
lq
y
i
BnX- An
r(BnX)__
(ki Bk-Ir(Bkl-Ix)
D
s
+. Bkir(Bkix)
for n -> i)
+
2Bki_ir(Bki_l x)
has a representation (15) and that every
s
K > 0
Then the proof of Theorem
a representation (15) holds for some
bounded by a constant which depends on
discrete (i.e.
i
1
D
Let
i
Then (note that
This and (6) show that every
number (15) belongs to
+
m.1
i
is finite since
K
Bn
and
....llBnXll
x
only.
<_
shows that for every
and
Hence, if
then
D(x)
which are
C(x)
is
is discrete
and vice versa.
2.
A slight modification of the proof of Theorem 1 also shows that
1/2
(n
lq)
implies
n/m
A /B
for some
([7]
13;[2]
W. B. JURKAT & A. PEYERIMHOFF
486
[4],
Theorem 184; for a more general result compare
k > i
choose
and
n <
B
n
such that
(-i)
If
)t _>
3.
THE STRUCTURE OF
k-I
LEMM 2.
If
x
C(x)
and
k
(n=l is a trivial case).
then (14) implies
n -<
)t < 1 since
C(x)
Bk,
WHEN
C(x)
We show first that
S
< n < B
then (14) leads to the contradiction
Be,
k.
O. But
C (x)
Bk_" I
hence
x
Proposition 4).
IS A
i, hence
-i/2
>
hence
IRRATIONALITY.
UADRATIC
is finite when
<
k
pAk_ 1.
m
lBk_ 1
(-i)
If
Inxl]
Inxll
0, >-n
p >
n
< 2n
0 <
In fact,
is a quadratic irrationality.
x
C(x)
belongs to a quadratic number field, then 0
and
are finite.
This Lemma is essentially due to Lekkerkerker
The following proof contains an explicit
[5],
see also Perron
[6], p.6.
representation of the elements of
c (x).
PROOF.
[b
x
o
tinued fraction, i.e.
i"’"
[b
x
=[pu P+I’"
,br_ i, po,
o
u
Pk-l’ Po "’Pu-I
It follows from (4) that
Pr+nk+v.-i
is represented in this case by a periodic con-
for
br+nk+v P
follows that
x
b
--Ip I’
Pn
’Pk ]
i
0,1,...,k-I
then
n e
and if
o
hence
(n
c
Pv-2’’’’’Po Pk-I ’’’’’p
lq
It
x
r+nk+v
ibn, bn-l"’"bl
k >- I
r > I
and the state-
/
ment of Lemma 2 follows from (3).
It follows from a theorem of Galois ([7], 23)
REMARK.
where
x
is the conjugate of
(-1)
X
x
if
k
C
Hence, the elements of
r+v-1
X
that
is even
This formula leads to an even more explicit
_+1
X
X
if
k
S
c
1
X
are
is odd.
(21)
representation of the elements of C S (x)
487
PROPERTIES OF QUADRATIC IRRATIONALS
This representation uses the notation
bj
]
bj+ I
([7]
5).
C
Then the elements of
(-I)
s(x)
x
([7],
Bi, j
19).
But
-
4.
+
[Po "’’’Pk-I
Bk-l’
if k is odd
D
+
(A__I
2
Bk_ 2) +
4(-1)
k-I
1,9-Bk-2,9 +
2
2Bk- I,
bjAi_l,j+ I
Ai, j
+
Bi_l,j+ I
+ 4(-1) k-I
([7], 5)
bv-l-2,v + Bk_2, v + -3, v bk- l+v-2, v + -3,v + Bk-2,v
-l,v-I +Bk-2,v-I
-l,v
denote the convergents of
(22)
Ai_l,j+ I
and it follows that
for the convergents of
if k is even
0, l,...,k-I
In fact, we have
n
n
n,j
are
Bk-l’9
rq-I
A /B
Let
/Bn, J
A
Hence
Bk-2,v
D
D
and (22) follows.
o
THE RELATION BETWEEN THREE CONSECUTIVE
Formula (8) shows that
lowing Lemma shows that
bn+ 1
bn+ 1
’s.
is a function of
is also a function of
n-l,
6
n,
n-l,
6
n+l"
alone.
n
The folThis fact
is the key to the following considerations, which will show that the converse of
Lemma 2 is also true.
LEPTA 3.
For
n
(n-I
bn+l
PROOF.
n+l
-
>- 0
0;
n
and
(6
n-l’
0;
a n)
I
Formulas (3), (6) and (8) imply
1
n
i
p
n
1
+ 1-46n6n _I
26
n
(6n-1,0;
n
(n >-0)
(23)
W. B. JURKAT & A. PEYERIMHOFF
488
and (23)follows from
(note that
=bn+l’ bn+2’’’" ]
[n+l
bn+l
is irrational).
n+l
REMARK.
n+l
Formulas (6) and (4) show that
Pn+l--bn+l’ bn’ bl]
(n
>
0)
and it follows
n>
O;
(n+l’
bn+l
n
(n+l’O;
I, b I > i.
The first formula (24) remains true for n
if
2
or
if
(
bn+ I
6
n
0.
i,
o
>
It follows from (8) that
by (I0) a function
(6n’ 6n-i
n-I
n+l X(n’ 6n-l)
unless
(6n-l’ 6n+l’
n
1
b
n
hence there exists
n -> 0
X(n’ n+l
unless
(26)
i, b
n
I
1
we find explicitely
1
46
The following theorem gives
Let
(25)
1/2
X(n, n-i
Using the function
THEOREM 2.
n > 0
n-1
such that
X
n+l
and similarly
exists such that
bn+l (n’ 6n+l
and the remark shows that also
n
(24)
1
n
Lemma 3 shows that a (universal) function
i.e. unless
n
n >-0
i
n
2n
P(
n’
n-
$n- I
i
(2 7)
in a more convenient form than Lemma 3.
k
n
1
Then
6
k
n-I # n (l-kn n
and
489
PROPERTIES OF QUADRATIC IRRATIONALS
k
n
(6 n,
bn+l
n
6
Assume that
Let
6n_l
k
n
+
k
b
Let
6n-I
n
e
n+l
[0,
6
n(l-kn6 n))
k
(k l-kn6 n)
n-I
I-6
(28)
k
n
n
I) which contradicts (23).
>
Then by (I0)and (28)
kn(l-kn6n)).
(0 0"6 n > (6 n 1
follows from Lemma 3
i >
n
(kn(l-knn )’ 1-8n
0;6n
(kn(l-kn6 n
>
n >_ I
which implies
since
0;6 n)
6_i
k
n
0.
kn (kn(l-kn6n)’
k -I
n
b
n+l
n
0;
>
(n_l,0;n)
>
-’(O,O;n)
>
(l-n’0;6n)
i
1
6n
1 > k
n
1
follows from Lemma 3
Figure 1 shows the areas of constancy for the function
5.
THE INFLUENCE OF
0
C(x).
Our next step is to introduce the assumption
n
for some
lq
LEMMA 4.
Let
%
C(x)
into our considerations.
0 < % <
n-I
I/
and
If
n > 1
4)
This interval is empty if
and if
0
n+2
k
n
are
I
> %
then
i.e.
n
>- %
and
Then, by
(28), (I0), (5)and (12)
and
n)
Then
6 k-1,
n n
26
n
n n
if
[0
n-I
2
+
1
26 k
I
kn(_l-kn6n)._
n-I
( n-l’ o; )=
(note that
6
6n_l
k
PROOF
if
> 0
4).
W. B. JURKAT & A. PEYERIMHOFF
490
(29)
n+l
n
Our proof depends on the inequality
PROOF.
(3O)
In order to prove (30) we observe that
I
2
z)
4z
+
%2
<
+
(2z-
%
and (30) follows from
z;z) -<
+
z
6n +6n+l
Assume that the assumptions of Lemma 4 hold and that
If
6
n->
1
b
n+l
>
6n
hence
> %
X <
_<
q(%,
-6 ;6
n n
-<
(6 n’
6n+2 6n+l
<
(,6 n
6n+l
-< (%,
6n+l
i-%-
6
n+l
<
I
> 1
REMARK.
If
n+l;6 n
6n+1
bn+2
bn+2
hI/1-
1
Similarily, if
but
>-
1
then by (8), (I0) and (30)
Jl-%-
bn+ 1 --(6n_l, 6n+l;6n)<
but
+ %)
2z
Formula (5) is for
for all
n
n > 2
(29)
a special case of
then it follows form (29) that
2)t <
-,
i/.
Lemma 4 will be used now
to show that the points
distance from the discontinuities of
if
0
(6n, 6n-1
C(x)
keep a certain
We introduce the notation
491
PROPERTIES OF QUADRATIC IRRATIONALS
q
and we assume chat
Let
1/2
n
n
k (l-k
n n
n
n
> I > 0
for some fixed
I > 0
for some
+
nE.
Formula (8) and Theorem 2imply
n > 2
2
and all
nn
k
(31)
2
6nkn-26n
2k
2 nkn
In what follows we need the inequality
and the formulas
Let
8
>
n-i
qn
1-46 nn
(2
k-I)
2
n
+I
>
(note that
(i-28)
n
kn
>
2
Then it follows from (31) that
/b
hence (use
2
k
i-4 8 (I-8)
n
n
nn
n n-I
n n+l
E_ <
>
8.n+1_<
2
8
1+
(a-b) /
(- + V’b
q
n+1
8nn+ I
n
46n6n_ 1
n-1
+ (28 nkn-1)
nn-Sn_ 1
1/3
It follows that
6
Let
6
n-i
> q
q
n-1 < n
(32)
6
Then it follows from (31) that
n
8n6n+ 1
hence, by Lemma 4
n_l-qn
26nkn- I+ QI-4 6n6n_ 1
It follows that
(n + 8n+l)
I-4 8n6n+l+(l-2n
1
2)
W. B. JURKAT & A. PEYERIMHOFF
492
(33)
Formula (4) implies that all points
(6 n’ 6 n-i
n > 2
are in a certain open
triangle, and some straight lines inside of this triangle are excluded by Theorem 2
(cf. figure I)
Fig. 1
Moreover, if
6
n
restriction for
> k > 0
(6n, n_l
then (29), (32) and (33) introduce some additional
To describe the remaining region we introduce the
following set.
Let
M(k)
perties
0 -<
<
I/,
denote the (open) set of points
(x, y) with the pro-
493
PROPERTIES OF QUADRATIC IRRATIONALS
x > %
y > %
and for
x <
(l-x
y <
(Figure 2 illustrates
x
+ y
<
1/2
or
M(%)
for
%
i/5 .)
Fig. 2
If
n
0
> %
and (33).
for all
n c I
then
(n, n-i
e
M(%)
for
n > 3
by (29), (32)
The combination of this result with the results of section 4 leads im-
mediately to
THEOREM 3.
such that
bn+l
There are (universal) functions
(n’ n-i
n+l X(n, n-i
and
n >-0
X
defined on
M(0)
494
W. B. JURKAT & A. PEYERIMHOFF
The functions
n
6.
>
and
( <
> 0
i/)
for all
LE
THE CONVERSE OF
q
n
then (
% > O.
M(%)
are continuous on every
X
M()
n- i
n
If
n -> 3.
for
2.
We use Theorem 3 to prove the following result of Lekkerkerker [5].
THEOREM 4.
C (x)
s
If
C (x)
0
is finite and
then
s
x
belongs to a
quadratic number field.
PROOF.
pairs
Let
(i’ j)
M()
If
a
for every
nk_l
/
.j
(i’ j)
(n’ n-i
a’
then
(X(i, j)
6nk+l X(nk,
the successor of a.
{(x,y)
U(a,e)
u(a,)
M()
(0,
N e 1
x
be so large that
depending on
n > N
A
since
nk
by Theorem
is finite, hence if
< e
a
a e A
Choose
U(a,e) n U(b,)
which is
F(n+l, n
(n+l’ 6n)
since
n > N.
a’
( n’
n
F( n
U(a,e
n-1
i.e. if
n-i
c_ U(a’,)
>
0
such that
# b
U(a’ e)
U(a,e
successor preserving"
(X(n’ n_l ),
then one
A
This establishes a mapping
Indeed, if
A
a
U(a,s)
a
(x,y)
if a
a
(34)
for exactly one
a
F(
n
n -> N
and
n
a
A
a
,i
A
The set
such that for every
((x,y)
Let
C(s)
0
Since
for all large
nk-l)--> (i’ j)
is constant on every
,
e
(x,y)
for every
It follows that
Choose
I
M()
be the set to all
A
a
of its later successors is again
Let
A
and let
on a subsequence.
/
such that
e A
implies
a’
We call
n-I
(n’
> 0
(i’ j)
C(x)
denote the elements of
with
there is some
a
.
F( n
i.e.
(n
by (34), hence
for every
n-i
then
a
n-i
e
n-i
(n+l’
n
a e A
,
U(a,e
U(a’,
then
PROPERTIES OF QUADRATIC IRRATIONALS
n > N
Take a fixed
cessors
a
and let
(o)
a
F(n, n_l).
a
a" ,...,a (1)
a
1
e lq
495
Consider a sequence of suc-
with
a
(1)
a.
It follows
that
F(6n++k 6n_l++k)
Since
a
()
Y is constant on every
U(a,e
bn++k+l
(Bn+9+K,Bn++k/_l)
fraction for
x
REMARK.
x
(i
+
,-i, k
0, i
0, I, 2,...
(35)
it follows from (35) that
is independent of k, i.e. the continued
This proves Theorem 4.
is periodic.
As conclusion we explain our results in the simplest case
/2
,i
(22), and D(x)
C(x)
Here
consists of the single point
1%2
consists of the points
(%,) # (0,0) by (16).
21/
%
i/by
with integral
It is well-known (see [3], p. 554) that
represents exactly the integers for which the exponents in the prime factor-
ization must be even for all primes
D(x)
__i
5
4__
2 or 3 nod 5
9__
Since this set contains only one element
Furthermore, given
C(y)
i/}
given by integral transformations
I__I
1__6
ax+b
cx+d
1__9
ad
y
y
with quotients
ACKNOWLEDGEMENT.
Foundation.
has period
1
1
C(x)
uniquely.
which produce this set are
bc
This follows because the proof of Theorem 4 works with
fraction for
20
(0,I) it determines
all possible
Y
So
+I.
i, so the continued
(the terms before the period being of no influence
by (22).
This research was supported in part by
the National Science
496
W. B. JURKAT & A. PEYERIMHOFF
REFERENCES
i.
Quadratic Irrationals in the Lower Lagrange
Davis, N. and Kinney, J.R.
Spectrum. Can. J. Math. 25 (1973) 578-584.
2.
Hardy, G.H. and Wright, E.M.
Oxford 1954.
3.
Hasse, H.
4.
Jurkat, W.B.
5.
Una Questione di Approssimazione Diofantea e Una Proprieta Caratteristica dei Numeri Quadratici I, II.
Atti Accad. Naz. Lincei. Rend. CI. Sci. Fis. Mat. Nat. (8) 21 (1956)
An Introduction to the Theory of Numbers.
Zahlentheorie, Akademie-Verlag, Berlin 1969.
Kratz, W., and Peyerimhoff, A.
of Dirichlet Approximations.
Explicit Representations
Lekkerkerker, C.G.
179-185, 257-262.
6.
Perron, O.
7.
Perron, O.
1954, 1957.
Uber die Approximation Irrationaler Zahlen Durch Rationale.
S.-B. Heidelberger Akad. Wiss. Math.-Nat. KI. 12 (192 I) 3-17.
Die Lehre von den Kettenbruchen.
Teubner Verlag, Stuttgart