INTEGERS 10 (2010), 65-71
#A6
VOLUME AS A MEASURE OF APPROXIMATION FOR THE
JACOBI-PERRON ALGORITHM
Fritz Schweiger
Department of Mathematics, University of Salzburg, Salzburg, Austria
fritz.schweiger@sbg.ac.at
Received: 11/7/08, Accepted: 11/24/09, Published: 3/5/10
Abstract
We consider the values of the consecutive minima of the quantities Fj (x; g) =
!d
(g+d+1)
(g+j)
(g+j) −1
(A0
+ j=1 A0
yj )(A0
) , 1 ≤ j ≤ d. W. Schmidt, in 1958, calculated the first and second minimum for j = 1 and d = 2. Schweiger, in 1975,
considered the case j = 1 for any d ≥ 2. This note is a continuation of these
investigations.
1. Introduction
W. Schmidt opened a new route on Diophantine approximation by the JacobiPerron algorithm when
" he introduced# volume as a measure of approximation. For
g ≥ d + 1, let p(g) =
(g)
(g)
A1
(g)
A0
,...,
Ad
(g)
A0
be the rational approximation to the point
x = (x1 , . . . , xd ) provided by the Jacobi-Perron algorithm. Then d consecutive
points p(g+1) , ...., p(g+d) , and x form a simplex with volume (y = T g x)
1
V (x; g) =
(g+1)
d!A0
(g+d)
...A0
(g+d+1)
(A0
+
d
!
j=1
.
(g+j)
A0
yj )
The Jacobi-Perron algorithm can be described by iteration of the map T on the
d-dimensional unit cube as follows (see [4]):
"
#
x2
1
T (x1 , . . . , xd ) =
− k1 (x), . . . ,
− kd (x)
x1
x1
$
%
$ %
xj+1
1
kj (x) =
, 1 ≤ j ≤ d − 1, kd (x) =
,
x1
x1
k(x) = (k1 (x), . . . , kd (x)).
The points x and z are called equivalent if there are n ≥ 0, m ≥ 0 such that
T n x = T m z.
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We further introduce the sequence k(g) (x) = k(T g−1 x) and the matrices

(g)
kd
1
(g)
k1
..
.



(g)
β (x) = 









(g+d+1)
A0
(g+d+1)
A1
..
.
(g+d+1)
Ad
(g+1)
(g)
kd−1
(g+d)
A0
(g+1)
A1
..
.
...
...
..
.
A0
(g+d)
A1
..
.
(g+1)
...
Ad
Ad

0 1
0 0 


0 0 
.. .. 

. . 
1 0
0 ...
0 ...
1 ...
.. . .
.
.
0 ...
(g+d)



 = β (1) (x) ◦ · · · ◦ β (g) (x).


In this note we consider the quantities
(g+d+1)
Fj (x; g) =
A0
+
d
!
k=1
(g+j)
A0
(g+k)
A0
yk
, 1 ≤ j ≤ d.
Let ξ > 1 be the largest root of X d+1 − X d − 1 = 0. In [2] the following
conjecture was stated. For all x with non-terminating expansion there are infinitely
many values of g such that the inequality
Fj (x, g) > ξ d+1−j + dξ −j
is satisfied.
Since for x∗ = ( 1ξ , ξ12 , ..., ξ1d ) it is easy to see that lim Fj (x∗ , g) = ξ d+1−j + dξ −j .
g→∞
This result was thought to be best possible.
For d = 1 this conjecture is true by Hurwitz’
famous result on continued fraction.
√
(Note that for d = 1 we have ξ + ξ −1 = 5.)
W. Schmidt [1] proved the conjecture for d = 2 and j = 1. For infinitely many
g ≥ 1, the inequality
1 1
F2 (x∗ , g) > 3ξ − 2 = lim F2 (( , 2 ), s) ∼ 2, 39671 . . .
s→∞
ξ ξ
is true. Moreover, he showed that if x is not equivalent to x∗ = ( 1ξ , ξ12 ), then the
constant ξ 2 + 2ξ −1 could be replaced by the greater value γ ∼ 4.26459 . . . which
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INTEGERS: 10 (2010)
is related to z ∗ = ( η1 + η12 , η1 ) where η 3 − 2η 2 − 3η = 1, η > 3. Again, this result
is best possible in an obvious sense. If x is not equivalent to x∗ or z ∗ then we
obtain
13
3
F1 (x, g) >
for infinitely many values of g.
Schweiger [3]proved that the conjecture is true for any d ≥ 1 and j = 1. Schweiger
[2] additionally proved that the conjecture is true for d = 2 and j = 2. In this
paper this matter is further explored. For d = 2 and j = 2 the second minimum is
calculated. Surprisingly the second minimum of F2 (x, g) is given by y ∗ = ( λ1 , λ1 + λ12 ),
λ3 = 2λ2 + 1, and not by z ∗ = ( η1 + η12 , η1 ) as for j = 1. Furthermore, it is shown
that the conjecture is not true for d = 3 and j = 3.
2. The Second Minimum
Theorem. Let λ > 1 be the greatest root of λ3 − 2λ2 − 1 = 0. Then for all x which
are not equivalent to ( 1ξ , ξ12 ) for infinitely many g ≥ 1 we have
F2 (x, g) > 3λ − 4 = lim F2
s→∞
""
1 1
1
, + 2
λ λ λ
#
#
, s ∼ 2.61671 . . . .
Proof. Here and in the sequel, overlines refer to a periodic expansion. We first
consider two special cases:
"
# " #
1 1
1
1
Case 1.
, +
=
, α3 − α2 − α − 1 = 0. Then
1
α α α2
lim F2
s→∞
Case 2.
"
""
#
#
, s = −α2 + 4α − 1 ∼ 2.97417 . . . .
1
2
#
, β 3 − 2β 2 − β − 1 = 0. Then
1 1
1
, + 2
β β
β
#
#
, s = −β 2 + 5β − 3 ∼ 3.24781 . . . .
1 1
1
, +
α α α2
1 1
1
, + 2
β β
β
lim F2
s→∞
""
#
=
"
(g+2)
Now consider F2 (x, g + 2) = k2
(g+2)
+ x2
If F2 (x, g) < 2.62 for all g ≥ g0 , then clearly
that g0 = 1, so that we have
1
3.
(g+2)
x2
=
(g+3)
+x1
k2
+x2
(g+3)
A0
(g+4)
A0
(t)
k2
(g+3)
k1
(g+3)
+
(g+3)
≥
(g+2)
(k1
(g+2)
+ x1
)+
(g+2)
A0
(g+4)
A0
.
≤ 2. Clearly, we may assume
1
9
(g+2)
and x1
=
(g+3)
k2
1
(g+3)
+x2
≥
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INTEGERS: 10 (2010)
(t)
(g+2)
Now let k2 = 2 infinitely often. Assume that k2
If
(g+2)
k1
= 2,
(g+4)
then clearly A0
(g+3)
≤ 2A0
(g+2)
+ 3A0
= 2.
(1)
. Hence,
(g+3)
F2 (x, g + 2) ≥ 2 +
(g+2)
Next, assume that k1
(g+1)
k2
(g+1)
or k2
(g+1)
= 1, k1
(g+2)
1 A0
1
A
+
(2 + ) + 0(g+4)
9 A(g+4)
3
A0
0
(g+3)
(g+2)
(g+3)
(g+2)
=
19 7A0
+ 3A0
+
(g+4)
9
3A0
≥
19 7A0
+ 3A0
+
(g+3)
(g+2)
9
6A0
+ 9A0
≥
19 2
25
26
+ =
>
.
9
3
9
10
= 1. Looking at Case 2 and Equation 1 we may assume
(g+1)
= 2, k1
(g+1)
= 0, k2
(g+1)
= k1
= 1,
= 0. In any case we obtain
(g+4)
A0
(g+3)
≤ 2A0
(g+1)
+ A0
.
Then again
(g+3)
F2 (x, g + 2) ≥
=
(g+2)
19 A0
1
A
+ (g+4) (1 + ) + 0(g+4)
9
3
A0
A0
(g+3)
(g+2)
(g+3)
(g+2)
19 4A0
+ 3A0
+
(g+4)
9
3A0
19 4A0
+ 3A0
25
+
≥
.
(g+4)
(g+1)
9
9
6A0
+ 3A0
" # " # " #
0
1
0
(g+2)
Now assume that k1
= 0 and note that only the digits
,
,
2
1
1
must be considered. Hence, the remaining case is
≥
(g+1)
k2
(g+1)
(g+1)
= 1, k1
=0
(the case k1
= 1 is not allowed by Perron’s condition for the digits).
(g+4)
(g+3)
(g+1)
We have A0
= A0
+ A0
, and we estimate
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INTEGERS: 10 (2010)
(g+3)
F2 (x, g + 2) ≥
(g)
(g+2)
(g+3)
(g+2)
19 A0
1 A
19
A
+ 3A0
+ (g+4) + 0(g+4) =
+ 0(g+3)
.
(g+1)
9
3
9
A0
A0
3A0
+ 3A0
(g)
(g)
(g)
(g)
Since k2 = k1 is not allowed, the cases k2 = 1, k1 = 0 and k2
remain.
(g)
(g)
(g+3)
(g+2)
(g)
If k2 = 1, k1 = 0, then A0
= A0
+ A0 , so that
(g+3)
(g+2)
+ 3A0
A0
(g+3)
3A0
(g)
+
(g+1)
3A0
(g+2)
=
4A0
(g+2)
3A0
+
(g)
+ A0
(g+1)
3A0
(g)
+
(g−1)
=0
5
.
9
≥
(g)
3A0
(g)
= 2, k1
(g−1)
If k2 = 2, k1 = 0, then we may assume that k2
= 2, k1
= 0. Calculation
(g+2)
(g+1)
(g−1)
(g+3)
(g+1)
(g)
(g−1)
gives A0
= 2A0
+ A0
and A0
= 4A0
+ A0 + 2A0
, so that
(g+3)
(g+2)
+ 3A0
A0
(g+3)
3A0
However
"
19
9
+
5
9
=
+
24
9
Finally, the case
#
0
.
1
(g+1)
3A0
=
(t)
k2
8
3
(g+1)
=
(g)
10A0
(g+1)
15A0
(g−1)
+ A0 + 5A0
+
(g)
3A0
+
(g−1)
6A0
≥
5
.
9
> 2.61671 . . . .
= 1 for all t ≥ t0 leads to the periodic cases
"
1
1
#
and
!
Remark. The point
1
x = (λ, ) = lim
g→∞
λ
,
(g+3)
A0
(g+2)
A0
(g+1)
,
A0
(g+2)
A0
-
lies in every triangle spanned by three successive points
g + 1, g + 2, g + 3.
equation
"
(j+2)
A0
(j+1) ,
A0
#
, j =
(j+1)
(j)
A0
A0
Furthermore, this point lies on the straight line with the
1
1
+
= 3λ − 4.
λ λ2
Therefore there are infinitely many values g such that
x1 + x2
(g+3)
A0
(g+2)
A0
(g+1)
+
A0
(g+2)
A0
1
1
+
> 3λ − 4.
λ λ2
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INTEGERS: 10 (2010)
Remark. For F1 (x, g) the second minimum is given by the point
"
#
1
1 1
0 0
( + 2, ) =
2 1
η η η
where η 3 − 2η 2 − 3η = 1, η > 3.
For F2 (x, g) this expansion gives two points of accumulation:
1
1 1
η+1
lim F2 (( + 2 , ), 2s + 1) =
(−3η 2 + 9η + 5) ∼ 1.83445 . . .
η η η
η2
s→∞
and
1
1 1
2η + 1
lim F2 (( + 2 , ), 2s) =
(−3η 2 + 9η + 5) ∼ 3.21924 . . . .
s→∞
η η η
η2
Therefore this expansion is not related to the second minimum.
3. A Counterexample
The general conjecture about the first minimum of the quantities Fj (x, g) [2,4] is
not true.
Letting j = d = 3, we have
(g+4)
F3 (x, g) =
A0
(g)
(g+1)
+ x1 A0
(g)
(g+2)
+ x2 A0
(g)
(g+3)
+ x3 A0
(g+3)
A0
.
Let ξ 4 = ξ 3 + 1 and consider again


0
1 1 1
 
z = ( , 2, 3) =  0 .
ξ ξ ξ
1
Then
lim F3 (z, s) = ξ +
s→∞
3
= 4ξ − 3 ∼ 2.52112 . . . .
ξ3
But if we consider λ > 1, the greatest root of λ4 = 2λ3 + 1, then the expansion of
 
0
1 1 1
 
w = ( , 2, 3) =  0 
λ λ λ
2
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INTEGERS: 10 (2010)
gives the smaller value
lim F3 (w, s) = λ +
s→∞
3
= 4λ − 6 ∼ 2.42768 . . . .
λ3
Acknowledgment The author wants to express his thanks to Karl Fuchs who did
the numerical calculations with help of Mathematica! .
References
[1] Schmidt, W., Flächenapproximation beim Jacobialgorithmus. Math. Ann. bf 136 (1958), 365–
374.
[2] Schweiger, F., The Metrical Theory of Jacobi–Perron
Berlin/Heidelberg/New York: Springer Verlag, 1973.
Algorithm.
LNM
334,
[3] Schweiger, F., Der Satz von Hurwitz beim Jacobialgorithmus. Monatsh. Math. 81 (1975),
215–218.
[4] Schweiger, F., Multidimensional Continued Fractions. Oxford: Oxford University Press, 2000.