nternat. J. Math. & Math. Sci.
Vol. 5 No. 3 (1982)609-612
609
SHORT PROOFS OF THEOREMS OF LEKKERKERKER
AND BALLIEU
MAX RIEDERLE
Eberhardstr. 14
79 Ulm/Donau
West Germany
(Received October 16, 1981)
For any irrational number
ABSTRACT.
points of
{z:
z=q(q- p), p
,
q
-
let A() denote the set of all accumulation
{0},
p and q relatively
prime}.
In this
paper the following theorem of Lekkerkerker is proved in a short and elementary way:
The set A() is discrete and does not contain zero if and only if
is a quadratic
The procedure used for this proof simultaneously takes care of a
irrational.
theorem of Ballieu.
KEY WORDS AND PHRASES. Lekkerkerker s Theorem, Approximation of numbeJ
I atio na
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
1.
Quadratic
10F05, 10F35.
INTRODUCTION.
This paper is easily readable by anyone familiar with the elements of continued
fractions, as far as
irrationals.
Lagrange’s
theorem on the periodic representation of quadratic
Throughout this paper,
denotes an irrational number which is repre-
sented by the regular continued fraction
-
[b0,bl,b 2
[bo,b I
and A() stands for the set of all the real accumulation points of {z:
p e
q
{0},
p and q relatively prime}.
n
and g > 0 set B(x,g)
(x
n
z
q(q-p),
Obviously, A() describes those
Dirichlet approximation qualities which occur infinitely often.
any sequence (a) let H(a
,bn_l,n]
Furthermore, for
denote the set of all its limit points and for x e R
g, x + g).
The main purpose of this paper is to gJ’e a
simple proof of the following theorem of Lekkerkerker
[i] (cf. also [2])"
M. RIEDERLE
610
A()
The set
is discrete and does not contain zero if and only if
is a
quadratic irrational.
The proof of the sufficient part of the theorem mainly depends on the irre-
,
ducible polynomial of
whereas the necessary part is a consequence of the relation
between A() and the sequence
(n)
and simultaneously establishes the following
theorem of Ballieu [3]:
The set H(
n
is finite and () is bounded if and only f
n
is a quadratic
irrational.
we shall show how to evaluate A() in
Finally for any quadratic irrational
an easy way.
BAS IC FORMULAS.
2.
In this section we state the formulas used in the sequel.
Let A /B
n
[bo,bl,b 2,... where An and Bn are relatively
6n Bn(Bn- An)" Then the following formulas
(-l)n
6
n
n+l + i/0n
the n-th convergent of
On
n
Bn/Bn_ 1
e]O:
and put
n
denote
prime.
Set
hold for all
(2.1)
n-i
-
PROOF.
6n_ I
(-)
On + I/n+I
(2.2)
n+l
i+i+46 n 6 n-i
n
(-I)
26 n
(2.3)
Equation (2.1) is an easy consequence of the well known identity
(-l)n/(Bn[Bnn+l_+ Bn_,l)),formula
An/Bn
using the identities
n
b
+
i/n+I
n
(2.2) can be derived from (2.1) when
b
and O
n
n
+ i/ 0n-i
and, finally
(2 3) can
be obtained when combining (2.1) and (2.2).
3.
PROOF OF LKKKERKERKER’S THEOREM.
(1)
Suppose
form f(x,y)
ax
2
+ bxy + cy
algebraic conjugate of
for all (p,q)
(
.
There exists an indefinite quadratic
is a quadratic irrational.
f(p,q)
x
2
with a,b,c e
.
and
f(,l)
O.
If
denotes the
then if follows by Vieta’s theorem that
a(p
q)(p
q)
aq(q- p)(
(.- {0}) which implies that
p/q)
(3.1)
611
THEOREMS OF LEKKERKERKER AND BALLIEU
f(P,q)
q(q- p)
{(x,y):
When using the notation
(3.2)
a(- p/q)
.,
x,y
y # 0, x and y relatively prime}
equation (3.2) implies that
A()
(Pn’qn)
since if
f(Z)
Z with
nlmoo qn(qn
(ii)
distinct,
-< 1 for all n
H(n)
6.
A().
and hence it follows by (3.3) that 0
q0"
From equation (2.1) we can
A().
n
Therefore and since all the numbers
is a compact subset of
Clearly,
Now from (3.1) we can see that
Suppose that A() is discrete and 0
16nI
see that
nlmoo Pn/qn
A() then
p)
and we conclude that A($) is discrete.
f(p,q) # 0 for all (p,q) e
(3 3)
a(- E)
H(n is finite and
and (n) is bounded.
are
H(n).
A() and hence
0
n)
Therefore,
Now by (2.3) it is easy to see that H(
is finite
in order to complete the proof, it suffices to prove Ballieu’s theorem.
PROOF OF BALLIEU’S THEOREM.
4.
Suppose that
(i)
($n,)
It follows from the identity
< k for all n
bn
H(n)
set
z
H(n) is finite say H(n) {z I
m
lq such that
a
there
that
k
exists
[bn’bn+l’’’"
bn + i/k < < bn + i i/(k + i). Therefore, the
is bounded and
n
n
IN, and hence
n -is empty and we can find a number
> 0 such that
g
th
sets
B(zv,g)
are pairwise disjoint and contained in]R-Z.
Z put
Let l(z) denote the greatest integer not exceeding z and for z
I(z)
(z
l(z))
-i
Clearly,
n+l I(n
for all n
IN0"
Also the function I is
H(n) therefore I(H(n) H(n) and we can find a 6, 0 < 6 < g,
m}
{i
There exists a number
such that I(B(z,$)) = B(I(z),g) for all v
m
such that
B(z,) for all n > n o Therefore, when
B(Zl,) and
o
=I
we obtain by induction that
writing (P) for the p-th composition map of
continuous on
n
n
n
O
the
B(I(P)(zl)’6)
sequence ((P)(zl)), p
I(T
by
for all p
+ p e
(p)
(Zl))
O
,
e0"
Since
is periodic.
H(n)
(H(n))
and
From the identities
we conclude that the sequence (b n
Lagrange’s theorem,
(ii)
lq0,
n
O
+
p)
P
0’
H(n
bno
is finite,
l(n o
+
p)
is periodic and thus,
is a quadratic irrational.
The other direction of Ballieu’s theorem is an easy consequence of
i. RI EDERLE
612
Lagrange’s
5.
theorem.
CONCLUDING REkRKS.
The inclusion in (3.3) is actually an equality. In order to prove this, we
need the following well known theorem (cf.
[4], p. 22-23):
Let f(x,y) be an indefinite quadratic form with integer coefficients and let
Then for any pair (p,q) e Z
be one of its roots.
many relatively prime integers
and
nlm (qn
p)
Pn’ qn
such that
x(-{0})
f(pn,qn
there are infinitely
f(p,q) for all n
O.
In fact, this result combined with (5) and (6), leads to the following:
THEOREM.
Suppose that
is a quadratic irrational, say
.
indefinite quadratic form f(x,y) with integer coefficients.
the algebraic conjugate of
f(,l)
0 for some
Moreover, let E be
Then
A()
f(z)
f(l,0)(E- )
REFERENCES
i.
LEKKERKERKER, C.G.
2.
JURKAT, W.B. and PERERIMHOFF, A.
3.
BALLIEU, R.
4.
CASSELS, J.W.S. An Introduction to Diophantine Approximation.
Press, Cambridge, 1965.
5.
PERRON, O.
Una questione di approssimazione diofantea e una proprieta
caratteristica dei numeri quadratici I, II. Atti Accad. Naz. Lincei.
Rend. CI. Sci. Fis. [at. Nat. 21 (1956) 179-185, 257-262.
Characteristic approximat’ion properties of
quadratic irrationals, Intern. J. of Math. and Math. Sci. i (1978)
479-496.
Sur des suites de nombres lies $ une fraction continue rgulire,
Acad. Roy. Belg. Bull. CI. Sci. 29 (1943) 165-174.
Die Lehre von den
Kettenbrchen.
Cambridge Univ.
Teubner Verlag, Stuttgart, 1954.