Arithmetic Distributions of Convergents Arising from Jacobi-Perron Algorithm

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1 Arithmetic Distributions of Convergents Arising from Jacobi-Perron Algorithm Valerie Berthe, H Nakaa, R Natsui To cite this version: Valerie Berthe, H Nakaa, R Natsui Arithmetic Distributions of Convergents Arising from Jacobi- Perron Algorithm Inagationes Mathematicae, Elsevier, 25, V n <lirmm-5297> HAL I: lirmm Submitte on Oct 26 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés

2 Arithmetic istributions of convergents arising from Jacobi-Perron algorithm Valérie Berthé, Hitoshi Nakaa an Rie Natsui Abstract We stuy the istribution moulo m of the convergents associate with the -imensional Jacobi-Perron algorithm for ae real numbers in (, ) by proving the ergoicity of a skew prouct of the Jacobi-Perron transformation; this skew prouct was initially introue in [6] for regular continue fractions Introuction For an irrational number x, < x <, we enote by pn q n the n-th convergent of x, which is efine by the regular continue fraction expansion coefficients of x In 988, H Jager an P Liaret [6] stuie the istribution properties of the pairs (p n, q n ) moulo m These properties were originally consiere by P Szüsz in [6], an then by R Moeckel [7] who use the ergoicity of geoesic flows over the moular surfaces: more precisely, they prove that given any positive integer m 2, for aex, the sequence {(p n, q n ) : n } is equiistribute moulo m over the set {(p, q) Z 2 m : p, q = Z m}, where Z m := Z/mZ an where the notation p,, p k stans for the subgroup of Z m generate by the elements p,, p k To prove this property, H Jager an P Liaret consiere in [6] the group of 2 2 matrices with entries from Z m an eterminant ±, that is, {( ) } a b G(m) = : a, b, c, Z c m, a bc = ± It is possible to show that {( a ) : a Z m} generates G(m) This fact implies the ergoicity of a G(m)-extension (a skew prouct inee) of the continue fraction transformation The equiistribution property of {(p n, q n ) : n } moulo m is then an easy consequence of the iniviual ergoic theorem A natural extension of this skew prouct was then introuce in [3] to euce the istribution of the approximation coefficients associate with the with the support of ACI Jeunes chercheurs 2 Combinatoire es mots multiimensionnels, pavages et numération This work has been one uring the secon an the thir authors visit at LIRMM, Montpellier

3 continue fraction algorithm; these results were also extene to the so-calle S-expansions, in the sense of [4]; see also for connecte results [] an [] The aim of the present paper is to generalize these equiistribution results to the -imensional Jacobi-Perron algorithm Note that the -imensional Jacobi-Perron algorithm reuces to the regular continue fraction algorithm Let us start with the efinition of the Jacobi-Perron algorithm We fix a positive integer 2 Let X = [, ) be enowe with the Borel σ-algebra B We first efine the map T : X X by T (x) = T ((x, x 2,, x )) = ( x2 x [ x2 x ],, x x [ x x ] [ ]), x x for x = (x, x 2,, x ) X if x, an T (x) =, otherwise; (X, T ) is calle the -imensional Jacobi-Perron algorithm Notice that there exists a unique absolutely continuous invariant probability measure µ for T which is equivalent to the Lebesgue measure (see for instance [3]) We put for x in X with x k(x) = k () (x) = (k, k 2,, k ) = ([ x2 x ], [ x3 x ],, [ x x ] [ ]),, x if x =, we set k(x) = ; we similarly efine ( ) k (s) (x) = k (s), k(s) 2,, k(s) = k ( T s (x) ) for s We then associate (x, x 2,, x ) with the column vector an consier the following matrix k k 2 P = () k x Then T ((x, x 2,, x )) correspons to P To construct the sequence x {( ) } p (k) q,, p(k) : k (k) q (k) of simultaneous approximation convergents of x from the -imensional Jacobi- Perron algorithm, we first efine Q () as the ( + ) ( + ) ientity matrix x x 2

4 I + ; we then efine recursively Q (n) for n as k (n) Q (n) := Q (n ) k (n) 2 k (n) We thus set for n Let us observe that Q (n) = p (n ) p (n +) p (n ) 2 p (n +) p (n ) p (n ) p (n) 2 p (n) 2 2 p (n ) p (n +) p (n ) q (n ) q (n +) q (n ) q (n) p (n) k Q () = k 2 = P (2) k It is well-known that for any x = (x, x 2,, x ) X, lim n p (n) i q (n) = x i for i hols In this paper, we prove that for almost every x X the sequences of vectors {(q (n ), q (n +),, q (n) ) : n } an {(p (n), p(n) 2,, p(n), q(n) ) : n } are both equiistribute moulo m for any integer m 2 More precisely we put Z + m = {(α, α 2, α + ) Z + m : α, α 2, α + = Z m } an + + c m = Z m (the carinality of Z m ) One easily sees that c m = ϕ + (m) = {(a, a 2, a + ) {,, m} + : gc(a,, a +, m) = }, where ϕ + enotes the Joran totient function of orer + ; we thus have (see for instance [5] or []) c m = m + p m( p (+) ), (3) 3

5 where the notation p m stans in all that follows for the prouct over the prime numbers p that ivie m We then have the following: Theorem Let m 2 be a nonnegative integer For almost every x X an for any (α, α 2, α + ), we have Z + m { n N : (q (n ), q (n +),, q (n) ) (α, α 2, α + ) (mo m)} lim N N { n N : (p (n) = lim, p(n) N = c m = ϕ + (m) = m + p m ( p (+) ) 2,, p(n), q(n) ) (α, α 2, α + ) (mo m)} N To prove this theorem, we consier for a given integer m 2, the group G(m) efine in a similar way as in [6]: { SL( +, Z m ) if is even, G(m) = SL ± ( +, Z m ) if is o, where SL(+, Z m ) stans for the matrices with entries in Z m with eterminant, whereas SL ± ( +, Z m ) stans for the matrices with entries in Z m with eterminant ± Let us recall that (see for instance [] or [9]) that + + SL( +, Z m ) = m (+)2 ( p i ) = m (+)/2 ϕ i (m) i=2 p n Let C m enote the carinality of G(m) Since SL( +, Z m ) is a subgroup of SL ± ( +, Z m ) of inex 2 if is o an m 2, one thus gets m (+)2 + i=2 p n ( p i ) = m C m = (+)/2 + i=2 ϕ i(m) if is even or m = 2 2m (+)2 + i=2 p n ( (4) p i ) = 2m (+)/2 + i=2 ϕ i(m) if is o an m 2 We ientify Q () with the matrix with coefficients in Z m obtaine by reucing moulo m its entries Here we note that et Q () = or if is respectively even or o, which implies that Q () belongs to the group G(m), whatever may be the parity of We efine the map T m on X G(m) by T m (x, A) = (T (x), AQ () ); T m is sai to be a G(m)-extension of the map T We efine the probability measure δ m on G(m) by ( C m,, C m ) Then it is easy to see that µ δ m is an invariant probability measure for T m Our i=2 4

6 question is whether (T m, µ δ m ) is ergoic or not In Section 2, we show that the set of matrices of the form (2) (reuce moulo m) generates G(m) Then in Section 3, we prove the ergoicity of T m, from which we euce the following proposition an then Theorem (in the same way as in [6]): Proposition For ae x X an any A G(m), lim N Finally we have the following N { n N : Q(n) A (mo m)} = C m Corollary For ae x X an any a Z m lim N N { n N : q(n) a (mo m)} = m ϕ (gc(a, m)) gc(a, m) ϕ + (m) In all that follows, we simply enote by,,, m the elements of Z m if it is clear that the elements are in Z m accoring to the context In this case, one has obviously m = 2 Basic properties of G(m) We first efine Then it is well-known that Γ m = {A SL( +, Z) : A I + (mo m)} SL( +, Z m ) = Γ m \ SL( +, Z) eg, see G Shimura [5], p 2 From this property, it easily follows that SL ± ( +, Z m ) = Γ m \ GL( +, Z) We respectively say that a ( + ) ( + ) matrix with Z (or Z m )-entries of the form (2) is a J-P matrix, an that a matrix of the form () is a J-P matrix; a J-P matrix is the inverse of a J-P matrix In the sequel of this section, we show that the monoi generate by the set of J-P matrices with Z m -entries is equal to G(m): Theorem 2 For any B G(m), there exist J-P matrices A, A 2,, A s such that B = A A 2 A s For this purpose, we first nee some notation an some preliminary lemmas We put k k 2 (k, k 2,, k ) := ; k 5

7 in particular, We thus have = (,,, ) = (a, a 2,, a + ) = (a +, a, a 2,, a ), (5) where a, a 2,, a + are either elements in Z m or ( + )-imensional vectors with Z m -entries Let us notice that the matrices (k, k 2,, k ) are J-P matrices Lemma For any (α, α 2,, α + ),A s with Z m -entries such that Z + m, there exist J-P matrices A, A 2, (α, α 2,, α + ) = (,,, )A A 2 A s, where s epens on (α, α 2,, α + ) Proof of Lemma We efine the following natural orer on Z m by m (+) Let (α, α 2,, α + ) Z m We enote by α the element in Z m such that α >=< α, α 2,, α Let us prove by inuction on α (consiere then as an element in {,, m}) that there exists a finite number of J-P matrices,, t an (α,, α ) Z m such that (α, α 2,, α + ) t = (, α,, α ) If α =, then α, α 2,, α = Z m an there exist k,, k Z m such that i= k iα i + α + = We thus have (α, α 2,, α + ) (k, k 2,, k ) = (, α,, α ) Suppose now that α Since α, α 2,, α + α, there exist k,, k Z m such that i= k iα i + α + α We thus have (α, α 2,, α + ) (k, k 2,, k, ) = ( k i α i + α +, α,, α ) We can now conclue inuctively since i= k iα i + α +, α,, α = Z m Now we have (, α,, α ) ( α,,, ) (, α,,, ) (,,, α ) i= = (,,, ) 6

8 Since a J-P matrix is the inverse of a J-P matrix, we get the assertion of this lemma The following lemmas are essential an easily prove Lemma 2 For any (+)-imensional vectors with Z m -entries (a, a 2,, a + ), we have (a,, a i, a i, a i+,, a, a + ) (,,,,,, ) i (,,, + i,,, ) i (,,,,,, ) = (a,, a i, a +, a i+,, a, a i ) Lemma 3 We have (a,, a + ) (,,, ) (,,, ) (,,, ) = (a, a,, a, a + ) In particular, when is o (a,, a + ) [ (,,, ) (,,, ) (,,, )] = (a, a 2,, a, a + ) Proof of Theorem 2 Let us fix b b 2 b (+) b 2 b 22 b 2(+) B = b (+) b (+)2 b (+)(+) in G(m) We want to prove that there exist J-P matrices,, s such that B s = I +, which implies immeiately the esire result For that purpose, let us prove by inuction on that there exist J-P matrices,, s such that ) B := B s =, (6) i ( I B () where I is the ientity matrix Inee, if this property hols for =, then we obtain that there exist J-P matrices,, s such that B s = ± i 7

9 If is even, then all J-P matrices are of eterminant Thus the ( +, + )- entry of the right han sie is equal to If is o an the ( +, + )-entry of the right han sie is equal to, then by Lemma 3 we can reuce it to by application of J-P matrices In either case, we get the esire result It thus remains to prove the inuction property Let us first prove that it hols for = Since et B = ±, then b, b 2,, b (+) = Z m, an there thus exist J-P matrices,, s such that (b, b 2,, b (+) ) s = (,,, ) by Lemma Thus b () b () b () (+) B s = b () b () b () (+) We set b () b () B () = b () b () Since et B () = ±, then there exist k,, k Z m such that b () b () B s (k,, k ) = b () b () It remains to set s = (k,, k ) to conclue the proof of the inuction property for = Let us assume now that the inuction property hols for (if =, the proof is finishe); one thus euces that the eterminant of B () (efine in (6)) is equal to ± We set B () = b () b () (+ ) b () (+ ) b () (+ )(+ ) Let us ivie the inuction proof into two steps for clarity issues Step Let us first prove that we can fin J-P matrices s +,, s +t 8

10 such that B s+ s+t is equal to I g (+) (l+) g (+) ( ) g (+) ( )(l+) g (+) ( )( ) g (+) (+ ) g (+) g (+) l g (+) ( )(+ ) g (+) ( ) g (+) ( )l (7) for some l, l < + Accoring to the proof of Lemma, we can fin ( + ) ( + ) J-P matrices (k (),, k() ),, (k(t),, k(t) ) such that (b (),, b() (+ ) ) (k(),, k() ) (k(u),, k(u) ) = (,,, ) Now ( ) I B () (,,,,, ) = }{{} an one checks more generally that for v ( ) I B () ( ) I, (,,,,, ) (,,,,, ) }{{}}{{} v+ {}}{ (,,,,,,,, ) = I }{{}}{{} } {{ } v v+ One thus gets that if u +, then ( ) I B () (,,, k () }{{},, k() ) (k(2),,,, k (2) }{{} 2,, k(2) ) (k (u),, k (u),, } {{, ) } has the esire form (7) Now if u + 2, then using (5), one gets ( ) I = ( ) I, 9

11 an suitable insertions of such as ( ) I B () (,,, k () }{{},, k() ) (k(2),,,, k (2) }{{} 2,, k(2) ) (k ( +),, k ( +),,, ) (,,, k ( +2) }{{}}{{},, k ( +2) ) (k (2( +)),, k (2( +)) l+,,, ) (,,, k 2( +)+ }{{}}{{},, k 2( +)+ ) (k (u),, k (u) l+,, } {{,, k (u) } ( l++),, k(u) ) provie the esire form (7), which ens the proof of Step Step 2 By (5), B s + s +t l+ = I g (+) (l+) g (+) ( ) g (+) ( )(l+) g (+) ( )( ) g (+) (+ ) g (+) ( )(+ ) I = g g ( ) g ( +) g ( ) g ( )( ) g ( )( +) g (+) g (+) l g (+) ( ) g (+) ( )l We put g g ( ) G = g ( ) g ( )( )

12 Since the eterminant of G is equal ±, there exist k,, k Z m such that I g g ( ) g ( +) (,,, k }{{},, k ) g ( ) g ( )( ) g ( )( +) I = G By applying (5), we get I I = G G We thus have prove that there exist J-P matrices s +t+,, s+t such that I B s+ s +t = G It remains to apply Lemma 2; there thus exist J-P matrices s+t +,, s+ such that ) B s + s+ =, ( I+ B (+) where we put g 2 g 3 g ( ) g B (+) =, g ( )2 g ( )3 g ( )( ) g ( )

13 which conclues the inuction proof 3 Ergoicity of T m an proof of Theorem 3 Ergoicity Let us recall some funamental facts about Jacobi-Perron algorithm For an integer vector a = (a, a 2,, a ) with a i for i, we put X a = {x X : k(x) = a} Then we see that X a if an only if a i a for any i an a > For a finite sequence of integer vectors {a (l) = (a (l), a(l) 2,, a(l) ), l n} such that a (l) i a (l), i, an a(l) > for l n, we efine the cyliner set of rank n by X a () a (n) = {x X : k(l) (x) = a (l) for l n} A cyliner set X a () a is sai to be proper (or full) if T n (X (n) a () a(n)) = X It is easy to see that X a () a is proper if (n) a(l) i < a (l) for all l n an i The following is essential Lemma 4 For almost every x X there exists a sequence of positive integers n < n 2 < such that X k () (x)k (n i ) (x) is proper for any i This shows the exactness of the ynamical system (X, T, µ); the exactness means here that n= T n B = {, X} (µ-mo ) In particular, (X, T, µ) is ergoic an strong mixing We refer to F Schweiger [2] or [4] about the theory of Jacobi-Perron algorithm Now we will show the ergoicity of T m Theorem 3 The skew prouct (X G(m), T m, µ δ m ) is ergoic Proof For any non-empty cyliner set X a () a(n), we see from Proposition 2 in [4] that sup x X a () a (n) DT n (x) < ( + ) + inf x X a () a (n) DT n (x), (8) where DT n is the Jacobian of T n Suppose that M is a T m -invariant set of (µ δ m )-positive measure Since T m acts as T on the first coorinate, the ergoicity of T shows {x X : (x, A) M for some A G(m)} = X (µ-mo ) Thus there exists A G(m) such that (X {A}) M has positive (µ δ m )- measure We fix such a set A By the ensity theorem an Lemma 4, for a given sequence ε i there exists a sequence of proper cyliner sets W i of rank n i an B G(m) such that for all i (µ δ m )((W i {A}) M) (µ δ m )(W i {A}) > ε i (9) 2

14 an a () a (2) A a () a (ni) 2 a (2) 2 a (ni) 2 = B (mo m), a () a (2) a (ni) where (a (), a() 2,, a() ),, (a(ni), a (ni) 2,, a (ni) ) are sequences of integers which efine W i From (8) we see that (9) implies (µ δ m )(Tm ni (W i {A}) M) > ( + ) + ε (µ δ m )(Tm ni i (W i {A})) Since W i is proper an M is T m -invariant, we conclue that (X {B}) M = X {B} ((µ δ m )-mo ) From Theorem 2, for any C G(m) there exist J-P matrices A,, A s such that C = BA A s (mo m) with a (i) A i = a (i) 2 a (i) for i s Moreover we can choose A,, A s so that the corresponing cyliner set X a () a (s) a (i) with a (i) =, i s, is proper This means a (i) Tm s (X {B}) X {C} an so M = X G(m) ((µ δ m )-mo ) Thus we get the assertion of the theorem 32 Proofs We are now able to give proofs of Proposition, Theorem, an Corollary Proof of Proposition Let us recall that C m enotes the carinality of G(m) From Theorem 3 an the iniviual ergoic theorem, we have lim N N { n N : T m n (x, B) X {A}} = C m 3

15 for (µ δ m )-ae (x, B) In particular, it hols for (x, I + ) for µ-ae x Since T n m(x, I + ) = (T n x, Q (n) ), we get the assertion Proof of Theorem + For any (α, α 2,, α + ) Z m, we enote by N (α,α 2,,α + ) the number of elements in G(m) such that the ( + )th row is (α, α 2,, α + ) We will show that N (α,α 2,,α + ) = C m m, () where C m enotes the carinality of SL(, Z m ) or SL ± (, Z m ) if is even or o, respectively It is easy to see that N (,,,) = C m m () From Lemma, we note that there always exists D G(m) such that the + ( + )th row is (α, α 2,, α + ) for any (α, α 2,, α + ) Z m For any matrix E of the form, ED is of the form This implies α α α + N (α,α 2,,α + ) N (,,,) On the other han, for any matrix D of the form, α α α + D D is of the form which implies, N (α,α 2,,α + ) N (,,,) Thus we have () From Proposition together with (), we have { n N : (q (n ), q (n +),, q (n) ) (α, α 2, α + ) (mo m)} lim N N = C m m = for µ-ae x C m c m 4

16 Inee one easily checks accoring to (3) an (4) that Cm m c m hols Since µ is equivalent to the Lebesgue measure, this hols for ae x with respect to the Lebesgue measure If we consier the (+)th column, then the same argument shows the other equality This completes the proof of Theorem C m = Proof of Corollary For a given a Z m, let Γ a (m) enote the carinality of the subset of G(m) of matrices whose ( +, + )-entry is equal to a Let us first assume that a an m are coprime We then euce from () that Γ a (m) = (α,,α ): α,,α,a =Z m N(α,, α, a) = (α,,α ) N(α,, α, a) = m 2 C m Let us assume now that m is a power of a prime ivisor p of a One has gc(α,, α, a, m) = if an only if gc(α,, α, m) = Hence Γ a (m) = (α,,α ): α,,α,a =Z m N(α,, α, a) = (α,,α ): gc(α,,α,m)= N(α,, α, a) = m C m ϕ (m) It easily euce from the Chinese remainer lemma that the functions m Γ a (m), m ϕ (m), an m C m are arithmetic multiplicative function Hence one checks that Γ a (m) = C m ϕ (gc(m, a)) m 2 gc(m, a) It remains now to apply Theorem to obtain the result, that is, for ae x X lim N N { n N : q(n) a (mo m)} = Γa(m) C m = Cm m2 ϕ (gc(a,m)) C m gc(a,m) = m ϕ (gc(a,m)) gc(a,m) ϕ + (m) Remark Let F q enote the finite fiel of carinality q an let F q [X] be the set of polynomials with F q -coefficients We enote by L the set of formal Laurent power series with negative egree Since L is a compact Abelian group, there exists a unique normalize Haar measure m We can efine the Jacobi-Perron algorithm on L( for any ) In this case, m is invariant uner this algorithm P Suppose that (n),, P (n) Q (n) is the n-th convergent of (f Q (n),, f ) L For any R F q [X], it is possible to prove the following : for any A,, A, A + F q [X] such that A,, A, A +, R are relatively prime, { n N : (P (n),, P (n), Q (n) ) (A,, A, A + ) (mo R)} lim N N = c R for m -ae (f,, f ) L, 5

17 where c R is a constant epening only on an R The proof is essentially the same as that of Theorem of this paper We refer to K Inoue an H Nakaa [5] for the stuy of the rates of convergence for Jacobi-Perron algorithm over L an to R Natsui [8] for the L-version of Jager-Liaret s result in the case of continue fractions References [] D Barbolosi, Sur le éveloppement en fractions continues à quotients partiels impairs, Mh Math 9 (99) [2] A Broise an Y Guivarc h, Exposants caractéristiques e l algorithme e Jacobi-Perron et e la transformation associée, Ann Inst Fourier (Grenoble) 5 (2) [3] K Daani an C Kraaikamp, A note on the approximation by continue fractions uner an extra conition, New York Journal of Mathematics 3A (998) 69 8 [4] C Kraaikamp, A new class of continue fraction expansions, Acta Arith LVII (99) 39 [5] K Inoue an H Nakaa, The moifie Jacobi-Perron algorithm over F q (X), Tokyo J Math 26 (23) [6] H Jager an P Liaret, Distributions arithmétiques es énominateurs e convergents e fractions continues, Inag Math 5 (988) 8 97 [7] R Moeckel, Geoesics on moular surfaces an continue fractions, Ergoic Theory Dynam Systems 2 (982) [8] R Natsui, On the group extension of the transformation associate to nonarchimeean continue fractions, preprint [9] M Newman, Integral matrices, Acaamic Press, New York Lonon, 972 [] V N Nolte, Some probabilitic results on the convergents of continue fractions, Inag Math (NS) (99) [] J Schulte, Über ie Joransche Verallgemeinerung er Eulerschen Funktion, Results Math 36 (999) [2] F Schweiger, The metrical theory of Jacobi-Perron algorithm, Lecture Notes in Mathematics- 334, Springer-Verlag, Berlin-New York, 973 [3] F Schweiger, Ergoic theory of fibre systems an metric number theory, Oxfor: Oxfor University Press, 995 [4] F Schweiger, Multiimensional continue fractions, Oxfor: Oxfor University Press, 2 6

18 [5] G Shimura, Introuction to the arithmetic theory of automorphic functions, Kano Memorial Lectures, No Publications of the Mathematical Society of Japan, No ;Tokyo : Iwanami Shoten; Princeton, NJ : Princeton University Press, 97 [6] P Szüsz, Verallgemeinerung un Anwenung eines Kusminschen Satzes, Acta Arith 7 (962), 49 6 Valérie Berthé LIRMM, 6 rue Aa, F Montpellier, France berthe@lirmmfr Hitoshi Nakaa an Rie Natsui Dept of Math Keio University Yokohama, Japan nakaa@mathkeioacp r natui@mathkeioacp 7