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 ISBN 0-387-06388-9 Springer- Verlag New York - Heidelberg • Berlin This work si subject to copyright. All rights are reserved, whether the whole or part of the material si concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction yb photocopying machine or similar means, and storage in data banks. Under § 45 of the German Copyright Law where copies are made roCi other than private use, a fee si payable to the publisher, the amount of the fee to be determined by agreement with tile publisher. © by Springer-Verlag Berlin - Heidelberg .3791 Library of Congress Catalog Card Number 73-920I. Printed in Germany. Julius Offsetdruck: Beltz, Hemsbach/Bergstr. 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 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

The Metrical Theory of Jacobi-Perron Algorithm

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