The Metrical Theory of Jacobi-Perron Algorithm

134 Pages·1973·2.069 MB·English

by  Fritz Schweiger (auth.)

Lecture Notes ni
Mathematics
A collection of informal reports and seminars
Edited yb .A Dold, Heidelberg dna .B Eckmann, hcirJ(Z
334
Fritz Schweiger
Universit~t Salzburg, Salzburg/(~sterreich
ehT Metrical Theory of
Jacobi-Perron Algorithm
galreV-regnirpS
Berlin-Heidelberg- York New 1973
AMS Subject Classifications (1970): 10-02, 10A30, 10F10, 10F20, 10K10, 10K15, 10K99, 28A10,
28A70, 28A65
ISBN 3-540-06388-9 Springer-Verlag Berlin • Heidelberg • New York
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Preface
In these Lecture Notes a valuable monograph on the algebraic and arith-
metic aspects of Jacobi algorithm by L. Bernstein [I] appeared. This
book should be a counterpart to it. It covers almost all aspects of
Jacobi algorithm not touched in the beautiful work by L. Bernstein.
These are questions of measure theory, ergodic theory, dimension and
diophanti~e approximation. The alqorithm is treated as a model of an
f-expansion (see R~nyi [I]) where all difficulties of multidimensiona-
lity enter. There are included some General results on erqodic theory
and dimension theory.
I want to thank V m teacher W.M. Schmidt to whom I owe my introduction
to Jacobi algorithm and to L. Bernstein who persuaded me to write these
notes.
My thanks also go to Mrs. Millonig for her patient job of typing and
to Dr. Fischer who discovered a lot of errors in previous versions.
University of Salzburg F. Schweiqer
March 1973
Table of Contents
§ L Basic definitions 1
§ .2 Cylinders 8
§ .3 Increasing u-fields 15
§ ,4 Conditional expectations 17
§ .5 Ergodicity of the transformation 22
§ ~6 Existence of an equivalent invariant measure 23
§ ~ The ergodic theorem 28
§ .8 Kuzmin's theorem 34
§ ~ Convergence results 42
§I~ The Borel-Cantelli lemma of Schmidt-Philipp 49
§11. Some extensions of Kuzmin's theorem 54
§12. Outer measures 56
§13. Hausdorff measures 6o
§14. Hausdorff dimension 63
§15~ Billingsley dimension 65
§16. Comparison theorems 68
§iZ The main theorem of dimension theory of Jacobi algorithm 75
§18, Further results on Billingsley dimension 85
§19, Ergodic invariant measures 9o
§2o, Volume as approximation measure 94
§21. Proof of the conjecture for n=l and n=2 99
Appendix I . Appendix II lo5
References lo6
The metrical theory of
Jacobi - Perron al~orithm
Notation
is the integral part of @
B n-dimensional unit cube after a suitable set of
Lebesgue measure zero has been removed
n-dimensional Lebesgue measure
T the basic transformation as defined in § 1
the basic invariant measure (§ )6
N the natural numbers
i. Basic definitions
General references are Perron ]I[ and Bernstein [i].
We begin with some formal definitions. For n fixed we define
the following set of (n+l)x(n+l) matrices
I
OO'''Ol
iO OO
A --- Ol OO
o
00...I0
A g ....
O O ... 1 ag~
where a g = (agl,...,ag n) is an integral vector. One sees
det A = (- )I n
g
- 2 -
Furthermore we define
~ g A o A 1 ... A g_l , g > 1
= E (unit matrix)
o
Then we have
A for all g > O
~g+l = ~ g g
Denoting ~g = ((Ai(g+J))) , i,j = O ..... n
we see
A i(j) = 6 ij (Kronecker delta), i,j = O,...,n
The definition of the A. (g+J) is easy to be seen well
posed due to the special nature of A , more precisely
g
A. (g+l+j-l) = A. (g+J) for j = 1 ..... n
as an element of as an element of
g+l g
and
A. %('l+n" = O 1 < i < n
A (n+l) = 1
o
and
n
A.(g+ n+l) = A. (g) + Z A. (g+j)
1 1 j=l l agj
for O < i < n , g > 1.
We note
det = det ((A. (g+J))) = (- i) ng
g l
We now set
K = {x I O <_ i x <I}.
and
W = {x i I x = O}
Then we define the following mapping
T : K \W + K
T(x I ..... Xn) = (X~l - ' .... Xl )
- 3 -
Then s T is defined on
s-I
U wJ-T
recursively by the formulae
0=3
° K = T i
T j+l = T j T O < j < s-i
We now define
B =K 0 T-J W
j=O
Then T : B ÷ B
will be basic for all further developments. Note that s T
is defined on B for all s > O. Next we define the following
sequence of functions
k : B +N n
s o
x + ks(X)
where n N denotes the set of all vectors with nonnegative
o
integral components:
i x2
kl(X) = ([~] ..... )
ks(X) = kl(TS-I x) s ~ 1
For x ~ B the vector ks(X) = (ksl(X),...,ksn(X)) is in
fact a vector in n N Associating to k (x) the matrix
o " s
/ /0... 0 i )
As(X) : ( 1 O ksl
\
O ... 1 ksn
we can define for each x the functions A~s+J) (x)- as explained at the
beginning of this chapter.
Lemma i.i: ~ T -j W is a countable union of countable pieces of
j=O
hyperplanes.
Proof: W is a piece of the hyperplane I = x O. It is enough to show that
the intersection of a hyperplane E with K has a counterimage consisting
of countable pieces of hyperplaneso
~IYI + ~Y2 + "'" + °nYn + ~o = O
- 4 -
Each y = (yl,...,yn) e K has the at most countable counterimages
T-ly = x = (mnlYn, mn-l+Yn-l) for m
"'" ' mn + Yn = (m I ..... mn) with
O <_ i m <_ n m and 1 <_ n m as can be seen from the definition of T (in
case that Yi-i ~ Yn ' i = 2,...,n, only i m n < m is allowed). Substi-
tuting this we have
x2 x3 1
~i(~i - ml ) + ~2(X~l - m2 ) + ... + ~n(xl -- - m n) + = 0o O
This gives the equation of a hyperplane and T-IE consists of the inter-
sections of these hyperplanes with the regions
2 x
< 1 < m +i}~ K
{ml ~ X~l < ml+l ..... mn -- x~ n
Lemma 1.2: If TSx = y, x e B, the following relations hold:
A~ s+n+l) + Z n A!s+j)
j=l l Yj
i x = n
Ao(S+n+l) + j=IZ A(s+J)° Yj
Proof: By induction, s = O°
n
. (n+l) + Z (J)
~i Ai Yj
xi .......... j=l = Yi
n (j)
A (n+l) + Z A
o j=l o Yj
by the definitions of the i A (J) . We assume the formula proved for s and
will show its truth for s+l. ~/~ p~ ~ :
TSx = y, TS+Ix = Ty = z.
The definition of T gives:
yj = ks+l'~-i + Z~'l for 2 ! J ! n
ks+l, n + n z
1
Yl =
ks+l, + z
n n
From this we have (O < g < n)
- 5 -
= A(s+n+l) n A(s+j) ks+~j_ 1 + ZJ-! +
A (s+n+l) + ~ A (s+j) yj +
g j=l g g j=2 g ks+l, n + n z
+ A(S+l) 1 = 1 (A~ s+n+l) +
ks+l,n
g ks+l, n + n z ks+l, n + n z
n-i (s+l+j-l) A(S+l) A(s+n+l)
Z A g g n
j-l=l g ks+l, j-i + + z +
n-I A(S+l+j_l) = 1 (s+l+n+l) n A(S+l+j)
+ Z 4 + (Ag + Z zj )
j-1=1 g z j-l) ks+l ,n z n j=l g
Substituting this in the formula for s we get our result.
We now give the following definition:
A sequence ~ = (al,a2,...) of vectors a •I e N On will be called admissible
if the following conditions hold:
(i) 0 <_ asi <-- asn ' s > 1 i = l,...,n
(ii) 1 < a
-- sn
(iii) The relations (0 < t < i-l)
a . = a
sl sn
as+l,i_ 1 = as+l,n_ 1
as+t,i_ t = as+t,n_ t
imply
as+t+l,i-(t+l ) ~ as+t+l,n-(t+l )
if t+l < i, and
1 ~ as+t+l,n_(t+l )
if t+l = i.
Lemma 1.3: If the sequence (al,a2,...) is admissible, the sequence
(a2,a 3 .... ) is admissible.
Proof:Clear.
Lel~na i. : 4 Given an admissible sequence (b I ,b 2 ,b 3, ... ) the sequence
- 6 -
(al,...,ag,bl,b2,...) , g ~ n-l, is admissible if and only if
)e( (a I ..... ag) is ad~ilissible that means, is the beginning of at
least one admissible sequence
(~) (ag_n+2 ..... ag,bl,b 2 .... ) is admissible.
Proof: The "only if" part is clear by Lemma 1.3.
The conditions )i( and (ii) are satisfied for (al,...,ag,bl,b2 .... ) by
(e). The condition (iii) has only influence to at most the next n-i
vectors (as can be seen by the worst case i = n-i and t = n-2)o
L emma 1.5: For any x £ B the sequence (kl(X) , k2(x),...) is admissible.
Proof: It is enough to prove the relations for s = i.
(i) O < xi+ 1 < i, I x > 0 gives
O < Xi+l < __i 1 < i < n-I
I x I x
-- -- _
0 < [kli]< [kln ]
Hence
( ii ) Clear
kli + (Tx) i
(iii) xi+ 1 : kl n + (Tx) n 1 < i < n - 1
If kli = kln from 0 ~ xi+ 1 < 1 we have
(Tx) i < (Tx) n
and hence
k2,i_ 1 < k2,n_ 1
Using
= k2,i-I + (T2x)i_ 1
(Tx) i
k2n + (T2X)n
= k2~n-I + (T2X)n-i ....
(Tx)
k2n + (T2x) n
we have if k2,i_ 1 = k2,n_ 1 is valid too
(T2x)i_l < (T2X)n_ 1
and hence
k3,i_ 2 < k3,n_ 2