From p-adic numbers to p-adic words

Transcription

1 From p-adic numbers to p-adic words Jean-Éric Pin1 1 January 2014, Lorentz Center, Leiden

2 References I S. Eilenberg, Automata, Languages and Machines, Vol B, Acad. Press, New-York (1976). M. Lothaire, Combinatorics on Words, Cambridge Mathematical Library (1997). P. Ochsenschlager, Binomialkoeffizenten und Shuffle-Zahlen, Techn. Bericht, Fachbereicht Informatik, T.H. Darmstadt (1981). J. Berstel, M. Crochemore and J.-É. Pin, Thue sequence and p-adic topology of the free monoid, Discrete Mathematics 76 (1989),

3 Outline Part 1: From integers to words (1) The p-adic valuation (2) Subwords and binomial coefficients (3) The p-adic and pro-p topologies Part 2: Converging sequences (1) A first example (2) The Thue-Morse sequence strikes back Part 3: Mahler s theorem on interpolation series

4 Part I From integers to words

5 The p-adic valuation and the p-adic norm Let p be a prime number. The p-adic valuation of a non-zero integer n is v p (n) = max { k N p k divides n } By convention, v p (0) = +. The p-adic norm of n is the real number n p = p v p(n) Finally, the metric d p is defined by d p (x,y) = x y p

6 Examples Let n = 1200 = n 2 = 2 4 n 3 = 3 1 n 5 = 5 2 n 7 = 1 Let x = 512 and y = 12. Then x y = 500 = Thus d 2 (x,y) = 2 2 d 5 (x,y) = 5 3 d p (x,y) = p 0 = 1 for p 2,5

7 p-adic integers Some properties of the p-adic norm: xy p = x p y p x+y p max{ x p, y p } The ring Z p of p-adic integers is the completion of the metric space (N,d p ). It can also be defined as the inverse limit of the rings Z/p n Z.

8 Extension to the free monoid We denote by A the free monoid on the set A. Note that the map n a n defines a monoid isomorphism from (N,+) onto the free monoid a. Question. Is it possible to extend the p-adic norm and the p-adic metric to words in some natural way? Two (related) solutions. One is based on the extension of binomial coefficients to words, the other one relies on p-groups (the pro-p topology).

9 Subwords Let u = a 1 a n and v be two words of A. Then u is a subword of v if there exist v 0,...,v n A such that v = v 0 a 1 v 1 a 2 v n 1 a n v n. aaba is a subword of aacbdcac brcdbr is a subword of abracadabra office is a subword of coefficient pin is a subword of capobianco.

10 Binomial coefficients (see Eilenberg or Lothaire) Given two words u and x = a 1 a 2 a n, the binomial coefficient ( u x) is the number of times that x appears as a subword of u. That is, ( ) u = {(u 0,...,u n ) u = u 0 a 1 u 1 a 2...u n 1 a n u n } x

11 Some examples ( ) ( ) abab abab = 2 = 3 a ab Indeed abab = abab = abab. ( ) abab = 1 ba If a is a letter, then ( ) ( ) u a n = u a and = a a m ( ) abbabaab Exercise. =?? aab ( ) n m

12 An extension of the p-adic distance Let u and v be two words. Put { ( ) u r p (u,v) = min x x d p (u,v) = 1 r p (u,v) ( ) } v mod p x with the usual convention min = and p = 0. Then d p is an ultrametric: (1) d p (u,v) = 0 if and only if u = v, (2) d p (u,v) = d p (v,u), (3) d p (u,v) max(d p (u,w),d p (w,v))

13 The case of natural numbers Theorem (Consequence of Lucas theorem) Let n and m be natural numbers. Then r p (a n,a m ) = p n m p In other words, for 0 < k < p n m p, ( ) ( ) n m mod p k k and ( n p n m p ) ( m p n m p ) mod p

14 p-adic norm This leads to the following definitions { ( ) ( ) u v r p (u,v) = min x x x 1 d p (u,v) = r p (u,v) v p (u) = log p r p (u,1) u p = p v p(u) = 1 r p (u,1) = d p(u,1) } mod p

15 Three ways of computing binomial coefficients (1) Magnus transformation (2) Pascal s rule for words (3) Matrices

16 The Magnus transformation The Magnus transformation µ : Z A Z A, defined by µ(a) = 1+a and µ(b) = 1+b and extended by multiplication. Thus µ(1) = 1 and µ(abab) = (1+a)(1+b)(1+a)(1+b) = (1+a+b+ab)(1+a+b+ab) = 1+a+b+ab+a+aa+ba+aba+b +ab+bb+abb+ab+aab+bab+abab = 1+2a+2b+aa+3ab+ba+bb +aab+aba+abb+bab+abab More generally µ(u) = ( ) u x (Exercise) x x A

17 Pascal s triangle ( ) n = m ( ) n 1 + m 1 ( ) n 1 m

18 Extending Pascal s rule to words Let u,v ( ) A and a,b A. Then u (1) = 1, 1 ( ) u (2) = 0 if u v and u v, v ( ) u ( ) ua if a b vb (3) = ( ) ( ) vb u u + if a = b vb v ( ) uv (4) = ( )( ) u v (Exercise) x x x 1 x 2 =x 1 x 2

19 An exercise on matrices Verify that, for all words u,v {a,b}, 1 ( ( u u ) a) ab 1 ( ( v v ) ( 0 1 u a) ab 1 ( ) ( uv uv ) a ab ( b) 0 1 v b) = ( 0 1 uv ) b Applying this formula to u = abb yields =

20 Computing binomial coefficients with matrices Let a 1 a 2 a n be a word. The function τ : A M n+1 (Z) defined by 1 ( ) ( u u ) ( u ) ( a a 1 a 2 a 1 a 2 a 3... u ( 0 1 u ) ( u ) ( a 2 a 2 a 3... u ( u ) ( a τ(u) = 3... u ( u ) a n ) a 1 a 2 a n ) a 2 a n ) a 3 a n is a morphism of monoids: τ(u)τ(v) = τ(uv).

21 Computing binomial coefficients modulo p The function τ p : A M n+1 (Z/pZ) defined by τ p (u) τ(u) mod p is a morphism of monoids. Furthermore, the monoid τ p (A ) is a finite p-group, that is, a group whose number of elements is a power of p.

22 Languages and binomial coefficients (1) L 1 = = { ( ) u } u {a,b} 0 mod 2 ab (b+a ( b(ab a) b ) ) a a b a b a b a b The transition monoid of this automaton (syntactic monoid of L 1 ) is the dihedral group D 4 (of size 8).

23 Languages and binomial coefficients (2) ( ) u L 2 = {u {a,b} + aa a 1 2 ( ) u 0 mod 2} ab a,b b b a,b 4 a 3 The syntactic monoid of L 2 is also D 4.

24 Languages and binomial coefficients (3) { ( ) ( ) u u L 3 = u {a,b} 0 mod 2 and a b ( ) ( ) ( ) } u u u mod 2 ab aa bb The syntactic monoid of L 3 is the quaternion group Q 8 (of order 8).

25 a 5 b b 6 a b b 1 a 2 a a 4 a 3 b b a 8 b b 7 a

26 Languages recognized by a p-group A language whose syntactic monoid is a finite p-group is called a p-group language. Theorem (Eilenberg-Schützenberger 1976) A language of A is a p-group language iff it is a Boolean combination of the languages { ( ) } u L(x,r,p) = u A r mod p, x for 0 r < p and x A.

27 Separating word by p-groups Let u and v be two words of A. A p-group G separates u and v if there is a monoid morphism ϕ from A onto G such that ϕ(u) ϕ(v). Proposition Any pair of distinct words can be separated by a finite p-group. Proof. Wlog, we may assume that u is not a subword of v. Let L = { w A ( } w u) 1 mod p. Then u L since ( ( u u) = 1 and v / L since v u) = 0.

28 The pro-p metric Let x and y be two words. Put v p (x,y) = min { k there is a group of order p k+1 that separates x and y} d p(x,y) = p v p(x,y) v p (x) = v p (x,1) x p = p v p(x) Then d p is an ultrametric.

29 An equivalent metric Proposition d p is an ultrametric uniformly equivalent to d p. Theorem A function f : A B is uniformly continuous for d p iff, for every p-group language L of B, f 1 (L) is a p-group language of A.

30 Part II Converging sequences

31 Binomial coefficients (see Eilenberg or Lothaire) Given two words u and x = a 1 a 2 a n, the binomial coefficient ( u x) is the number of times that x appears as a subword of u. That is, ( ) u = {(u 0,...,u n ) u = u 0 a 1 u 1 a 2...u n 1 a n u n } x

32 Some examples ( ) ( ) abab abab = 2 = 3 a ab Indeed abab = abab = abab. ( ) abab = 1 ba If a is a letter, then ( ) ( ) u a n = u a and = a a m ( ) abbabaab Exercise. = 7 aab ( ) n m

33 An extension of the p-adic distance Let u and v be two words. Put { ( ) u r p (u,v) = min x x d p (u,v) = 1 r p (u,v) ( ) } v mod p x with the usual convention min = and p = 0. Then d p is an ultrametric: (1) d p (u,v) = 0 if and only if u = v, (2) d p (u,v) = d p (v,u), (3) d p (u,v) max(d p (u,w),d p (w,v))

34 The Magnus transformation The Magnus transformation µ : Z A Z A, defined by µ(a) = 1+a and µ(b) = 1+b and extended by multiplication. Thus µ(1) = 1 and µ(abab) = (1+a)(1+b)(1+a)(1+b) = (1+a+b+ab)(1+a+b+ab) = 1+a+b+ab+a+aa+ba+aba+b +ab+bb+abb+ab+aab+bab+abab = 1+2a+2b+aa+3ab+ba+bb +aab+aba+abb+bab+abab More generally µ(u) = ( ) u x (Exercise) x x A

35 An exercise on matrices Verify that, for all words u,v {a,b}, 1 ( ( u u ) a) ab 1 ( ( v v ) ( 0 1 u a) ab 1 ( ) ( uv uv ) a ab ( b) 0 1 v b) = ( 0 1 uv ) b Applying this formula to u = abb yields =

36 The pro-p metric Let x and y be two words. Put v p (x,y) = min { k there is a group of order p k+1 that separates x and y} d p(x,y) = p v p(x,y) v p (x) = v p (x,1) x p = p v p(x) Then d p is an ultrametric.

37 An equivalent metric Proposition d p is an ultrametric uniformly equivalent to d p. Theorem A function f : A B is uniformly continuous for d p iff, for every p-group language L of B, f 1 (L) is a p-group language of A.

38 Converging sequences Let (u n ) n 0 be a sequence of words and let u be a word. Are equivalent: (1) lim n u n = u, (2) For every k > 0, there is an N such that, for all n N and( for) all words ( ) x such that un u 0 < x k, mod p x x (3) For every k > 0, there is an N such that, for all n N and for all morphisms ϕ from A to a p-group of order p k, ϕ(u n ) = ϕ(u).

39 A first example of converging sequence Proposition For every word u A, lim n u pn = 1. Proof. By the definition of the topology, it suffices to show that if ϕ : A G is a monoid morphism onto a p-group G, then lim n ϕ(g pn ) = 1. But if G = p k, then for n k, ϕ(g pn ) = ϕ(g) pn = 1 since the order of ϕ(g) divides p k.

40 The pro-p free group The completion of the metric space (A,d p ) is denoted by F p (A). Since the product on A is uniformly continuous, it extends uniquely to F p (A). Also, for all u F p (A) ( ) 1 lim = 1 = u lim n upn n upn It follows that lim n u pn 1 = u 1 and hence F p (A) is a group, called the pro-p free group on A. It can also be defined as the inverse limit of the inverse system of all discrete finite p-groups.

41 Thue-Morse sequence Let A = {a,b}, and let τ : A A be the monoid morphism defined by τ(a) = ab and τ(b) = ba. For every n 0, let u n = τ n (a) and v n = τ n (b). Then for each i, u i is a proper left factor of u i+1. The sequence u 0, u 1,..., defines an infinite word t = abbabaabbaababba called the infinite word of Thue-Morse. u 3 = abbabaab v 3 = baababba

42 Ochsenschlager s theorem Theorem (Ochsenschlager 1981) For ( every word x such that 0 x n, un ) ( x = vnx ). Furthermore there exists a word x of length n+1 such that ( u n ) ( x vnx ). For instance, ( ) abbabaab = 7 aab ( ) abbabaab = 4 aaab ( ) baababba = 7 aab ( ) baababba = 2 aaab

43 The first values of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

44 The first values of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba n a b aa ab ba bb aaa aab aba abb baa bab bba

45 The first values of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba n a b aa ab ba bb aaa aab aba abb baa bab bba

46 The first values mod 2 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

47 The first values mod 2 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

48 The first values mod 3 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

49 The first values mod 3 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

50 The first values mod 3 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

51 The first values mod 3 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

52 The first values mod 3 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

53 The first values mod 3 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

54 The first values mod 3 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

55 The first values mod 5 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

56 The first values mod 5 of ( ) t[n] x for 0 < x 3 n a b aa ab ba bb aaa aab aba abb baa bab bba

57 The first values of f(2,n) n f(2, n) n f(2, n) n f(2,n)

58 Counting modulo 2 Consider the Fibonacci sequence defined by F 0 = 0, F 1 = 1 and F n+2 = F n+1 +F n for every n 0. Theorem (Berstel, Crochemore, Pin 1989) For every n 0 and for every word x such that 0 < x < F n, ( u n x ) ( vnx ) 0 mod 2. Corollary The sequence t[f(2,n)] converges to 1 in the pro-2 topology.

59 Counting modulo p 2 Let f(p,n) = 2 n p 1+ log pn. Theorem (Berstel, Crochemore, Pin 1989) For every prime number p and for every n > 0, there exists an m = f(p,n) such that, for every nonempty word x of length n, ( ) t[m] x 0 mod p. Corollary The sequence t[f(p,n)] converges to 1 in the pro-p topology..

60 Part III Mahler s theorem on interpolation series Mahler s theorem is the dream of math students: A function is equal to the sum of its Newton series iff it is uniformly continuous. s_theorem

61

62 References I K. Mahler, An interpolation series for continuous functions of a p- adic variable, J. Reine Angew. Math. 199 (1958), Correction 208 (1961), J.-É. Pin and P.V. Silva, A noncommutative extension of Mahler s theorem on interpolation series, European Journal of Combinatorics 36, (2014),

63 The difference operator Let f : N Z be a function. We set Note that ( f)(n) = f(n+1) f(n) ( 2 f)(n) = f(n+2) 2f(n+1)+f(n) ( k f)(n) = ( ) k ( 1) r f(n+r) r 0 r k

64 Newton series For each function f : N Z, there exists a unique family a k of integers such that, for all n N, f(n) = ( ) n a k k k=0 This family is given by a k = ( k f)(0) Thus the sequence of polynomial functions f m (n) = m k=0 a k( n k) pointwise converges to f.

65 Examples Fibonacci sequence: f(0) = f(1) = 1 and f(n) = f(n 1)+f(n 2) for (n 2). Then f(n) = ( ) n ( 1) k+1 f(k) k k=0 Let f(n) = r n. Then f(n) = ( ) n (r 1) k k k=0

66 Examples (2) The parity function f(n) = then f(n) = Factorial n! = { 0 if n is even 1 if n is odd ( ) n ( 2) k 1 k k>0 ( ) n a k k k=0 where the a k are derangements: number of permutations of k elements with no fixed points: 1,0,1,2,9,44,265,1854,14833,133496,

67 Mahler s theorem Theorem (Mahler) Let f(n) = k=0 a n k( k) be the Newton series of a function f : N Z. TFCAE: (1) f is uniformly continuous for the p-adic norm, (2) the sequence of polynomial functions ) f m (n) = m k=0 a n k( k converges uniformly to f, (3) lim k a k p = 0. (2) means that lim m sup n N k=m a k( n k) p = 0.

68 Mahler s theorem (2) Theorem (Mahler) f is uniformly continuous iff its partial Newton series converges uniformly to f. The most remarkable part of this theorem is the fact that a function is equal to the sum of its Newton series if and only if it is uniformly continuous.

69 Examples The Fibonacci function is not uniformly continuous (for any p). The factorial function is not uniformly continuous (for any p). The function f(n) = r n is uniformly continuous iff p r 1 since f(n) = k=0 (r 1)k( n { k). 0 if n is even If f(n) = 1 if n is odd then f(n) = k>0 ( 2)k 1( n k) and hence f is uniformly continuous for the p-adic norm iff p = 2.

70 Extension to words Is it possible to obtain similar results for functions from A to Z? Questions to be solved: (1) Extend difference operators to word functions. (2) Find a notion of Newton series for functions from A to Z. (3) Extend Mahler s theorem.

71 Difference operator Let f : A Z be a function. For each letter a, we define the difference operator a by ( a f)(u) = f(ua) f(u) One can now define inductively an operator w for each word w A by setting ( 1 f)(u) = f(u), and for each letter a A, ( aw f)(u) = ( a ( w f))(u)

72 Direct definition of w Example w f(u) = 0 x w ( 1) w + x ( w x ) f(ux) aab f(u) = f(u)+2f(ua)+f(ub) f(uaa) 2f(uab)+f(uaab) The following formula hold for all v,w A : vw f = v ( w f)

73 Newton series for word functions Theorem (cf. Lothaire) For each function f : A Z, there exists a unique family f,v v A of integers such that, for all u A, f(u) = ( ) u f,v v v A This family is given by f,v = ( v f)(1) = 0 x v ( ) v ( 1) v + x f(x) x

74 An example Let f : {0,1} N the function mapping a binary word onto its value: f(010111)= f(10111)= 23. { ( v f +1 if the first letter of v is 1 f) = f otherwise { ( v 1 if the first letter of v is 1 f)(ε) = 0 otherwise Thus, if u = 01001, then f(u) = ( ( u 1) + u ( 10) + u ) ( 11 + u ) ( u ) ( u = ) =

75 Mahler s theorem for word functions Theorem (Main result) Let f(u) = v A f,v ( u v) be the Newton series of a function f : A Z. TFCAE: (1) f is uniformly continuous for d p, (2) the sequence of polynomial functions ) f n (u) = 0 v n f,v ( u converges uniformly to f, (3) lim v f,v p = 0. v

76 Mahler s expansion of the product of two functions An interesting question is to compute the Mahler s expansion of the product of two functions.

77 Mahler s expansion of the product of two functions An interesting question is to compute the Mahler s expansion of the product of two functions. Proposition Let f and g be two word functions. The coefficients of the Mahler s expansion of fg are given by fg,x = v 1,v 2 A f,v 1 g,v 2 v 1 v 2,x where v 1 v 2 denotes the infiltration product.

78 Infiltration product (Chen, Fox, Lyndon) Intuitively, the coefficient u v,x is the number of pairs of subsequences of x which are respectively equal to u and v and whose union gives the whole sequence x. For instance, ab ab = ab+2aab+2abb+4aabb+2abab 4aabb since aabb = aabb = aabb = aabb = aabb 2aab since aab = aa b b = aab b ab ba = aba+bab+abab+2abba+2baab+baba

79 Infiltration product (2) The infiltration product on Z A, denoted by, is defined inductively by (u,v A and a,b A) u 1 = 1 u = u, { (u vb)a+(ua v)b+(u v)a if a = b ua bv = (u vb)a+(ua v)b if a b for all s,t Z A, s t = u,v A s,u t,v (u v)

80 Mahler polynomials A function f : A Z is a Mahler polynomial if its Mahler s expansion has finite support, that is, if the number of nonzero coefficients f, v is finite. Proposition Mahler polynomials form a subring of the ring of all functions from A to Z for addition and multiplication.

81 Ongoing research Mahler s theorem is about functions from N to N. We extended it to functions from A to N. Next step: functions from A to B.