A noncommutative extension of Mahler s theorem on interpolation series 1

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1 A noncommutatie etension of Mahler s theorem on interpolation series 1 Jean-Éric Pin2, Pedro V. Sila 3 Tuesday 20 th December, 2011 Abstract In this paper, we proe an etension of Mahler s theorem on interpolation series, a celebrated result of p-adic analysis. Mahler s original result states that a function from N to Z is uniformly continuous for the p-adic metric d p if and only if it can be uniformly approimated by polynomial functions. We proe the same result for functions from a free monoid A to Z, where d p is replaced by the pro-p metric, the profinite metric on A defined by p-groups. The aim of this paper is to gie a noncommutatie ersion of Mahler s theorem on interpolation series [8], which applies to functions from a free monoid to the set of integers. This result was first announced in [15] and this article is the full ersion of this conference paper. The classical Stone-Weierstrass approimation theorem states that a continuous function defined on a closed interal can be uniformly approimated by a polynomial function. In particular, if a real function f is infinitely differentiable in the neighbourhood of 0, it can be approimated, under some conergence conditions, by its Taylor polynomials k n=0 f (n (0 n! n The p-adic analogue of these results is Mahler s interpolation theorem [8]. It is is usually stated for p-adic alued functions on the p-adic integers, but this full ersion can be easily recoered from the simpler ersion gien in 1 Work supported by the project ANR 2010 BLAN FREC, the European Regional Deelopment Fund through the programme COMPETE and the Portuguese Goernment through FCT (Fundação para a Ciência e a Tecnologia under the project PEst- C/MAT/UI0144/ LIAFA, Uniersité Paris 7 and CNRS, Case 7014, Paris Cede 13, France. Jean-Eric.Pin@liafa.jussieu.fr 3 Centro de Matemática, Faculdade de Ciências, Uniersidade do Porto, R. Campo Alegre 687, Porto, Portugal. psila@fc.up.pt 1

2 Theorem 1 below. First, for each function f : N Z, there eists a unique family a k of integers such that, for all n N, f(n = ( n a k k k=0 or, in short notation, f = k=0 a ( k k. The coefficients ak are gien by the formula a k = ( k f(0 where is the difference operator, defined by ( f(n = f(n + 1 f(n. In other words, the binomial functions ( k form a basis of the Z-module of all functions from N to Z. The series k=0 a ( k k is called the Mahler s epansion of f. Mahler s theorem states that f is uniformly continuous for the p-adic metric if and only if the partial sums of its Mahler s epansion conerge uniformly to f. More precisely: Theorem 1 (Mahler Let k=0 a ( k k be the Mahler s epansion of a function f : N Z. The following conditions are equialent: (1 f is uniformly continuous for the p-adic metric, (2 the sequence of partial sums n k=0 a ( k k conerges uniformly to f, (3 lim k a k p = 0. The most remarkable part of this theorem is the fact that any uniformly continuous function can be approimated by polynomial functions, in contrast to Stone-Weierstrass approimation theorem, which requires much stronger conditions. This nice feature makes Mahler s theorem the dream of students in mathematics: a function is equal to the sum of its Newton series if and only if it is uniformly continuous. Our goal is to gie a noncommutatie etension of this result, based on the following obseration: since the additie monoid N is isomorphic to the free monoid on a one-letter alphabet, the free monoid A on a finite alphabet A can be iewed as a noncommutatie generalization of the natural numbers. Accordingly, our noncommutatie etension concerns functions from A to Z. Neertheless, generalizing Mahler s theorem to these functions requires a few preliminary questions to be soled: (1 Etend the difference operators to word functions, (2 Etend the binomial coefficients to words, (3 Find a metric on A which generalizes d p, (4 Find a Mahler epansion for functions from A to Z. Fortunately, (1 can be readily soled and (2 and (3 hae already be addressed by Eilenberg [5]: there is a well-defined notion of binomial coefficients for two words u and and the pro-p metric on A is a natural 2

3 etension of the p-adic metric. Question (4 is the topic of Section 2, which culminates with Theorem 2.2: just like in the commutatie case, eery function f : A Z has a Mahler s epansion of the form A f, ( where the coefficients f, are integers related to the difference operators. There is also a striking similarity between Mahler s theorem and our main result, which can be stated as follows: Theorem 2 Let A f, ( be the Mahler s epansion of a function f : A Z. The following conditions are equialent: (1 f is uniformly continuous for the pro-p metric, (2 the sequence of partial sums 0 n f, ( conerges uniformly to f, (3 lim f, p = 0. Although this result can be considered as interesting on its own right, the reader may wonder about our original motiation for proing this theorem. It was actually inspired by automata theoretic questions related to the classification of regularity-presering functions in language theory (see [14, 16] for more details. We discuss further this original motiation in Section 7. In Section 3, we introduce the notion of Mahler polynomials and proe some of their properties. Section 4, deoted to uniform continuity, is a crucial step towards the proof of the main theorem, presented in Section 5. In Section 6, we come back to the commutatie case by proiding an alternatie proof for Amice s Theorem, a generalization of Mahler s Theorem to functions from N k to Z. 1 Background and notations In this paper, p denotes a fied prime number. 1.1 The p-adic aluation and the p-adic metric The p-adic aluation of a nonzero integer n is the natural number { } ν p (n = ma k N p k diides n and its p-adic norm is the real number n p = p νp(n By conention, we set ν p (0 = + and 0 p = 0. Note that, for all, y Z, + y p ma{ p, y p } p + y p (1.1 y p = p y p min{ p, y p } (1.2 3

4 The p-adic metric d p on N and Z is defined by setting, for each pair (, y of integers, d p (, y = y p (1.3 These definitions can be etended to N k and to Z k as follows. Gien = ( 1,..., k and y = (y 1,..., y k in Z k and a positie integer r, we write y (mod r if, for 1 i k, i y i (mod r. The p-adic aluation of is ν p ( = min { n N 0 (mod p n+1 } and the p-adic norm is p = p νp(. Finally, the p-adic metric on Z k is defined by the formula (1.3, for each pair (, y of elements of Z k. It is well-known that d p is actually an ultrametric, that is, satisfies the following stronger form of the triangle inequality, for all u,, w in Z k : 1.2 Words and languages d p (u, ma{d p (u, w, d p (w, } Let A be a finite alphabet. Recall that the free monoid A on A is the set of all words on A, endowed with the concatenation operation. The empty word, denoted 1, is the identity of A. The length of a word u is denoted u. If a is a letter of A, we denote by u a the number of occurrences of a in u. A language is a subset of A. An important class of languages is the class of recognizable (also called regular languages. Recognizable languages admit arious characterizations, ranging from automata theory to topology or logic, but we will settle for the following algebraic definition: a language L of A is recognizable if and only if there eists a monoid morphism ϕ from A onto some finite monoid M such that L = ϕ 1 (ϕ(l. It is then said that M recognizes the language L. 1.3 Binomial coefficients Let u and be two words of A. Let u = a 1 a n, with a 1,..., a n A. Then u is a subword of if there eist 0,..., n A such that = 0 a a n n. For instance aaba is a subword of caacabca. Following [5, 7], we define the binomial coefficient of u and by setting ( u = {( 0,..., n = 0 a a n n }. Obsere that if a is a letter, then ( a is equal to a. Further, if u = a n and = a m, then ( ( m = u n 4

5 and hence these numbers constitute a generalization of the classical binomial coefficients. The net proposition, whose proof can be found in [7, Chapter 6], summarizes the basic properties of the generalized binomial coefficients and can sere as an alternatie definition. Lemma 1.1 Let u, A and a, b A. Then (1 ( u 1 = 1, (2 ( u = 0 if u and u, (3 ( {( u ua b = b if a b ( + u if a = b b A third way to define the binomial coefficients is to use the Magnus automorphism of the algebra Z A of polynomials in noncommutatie indeterminates in A defined by µ(a = 1+a for all a A. One can show that, for all u A, µ(u = ( u (1.4 A which leads to the formula ( u1 u 2 = 1 2 = ( ( u1 u2 1 2 ( The pro-p metric Let u and be two words of A. A p-group G separates u and if there is a monoid morphism from A onto G such that ϕ(u ϕ(. One can show that any pair of distinct words can be separated by a p-group [5]. The pro-p metric d p on A is defined by setting r p (u, = min { n there is a p-group of order p n+1 separating u and } d p (u, = p rp(u, with the usual conention min = + and p = 0. It turns out that d p is an ultrametric. One can also show that the concatenation product on A is uniformly continuous for this metric. It follows that the completion of the metric space (A, d p is naturally equipped with a structure of monoid. This monoid is in fact a compact group, called the free pro-p group and denoted F p (A. When A is a one-letter alphabet, this group is of course Z p, the additie group of p-adic integers. One can define in a similar way a pro-p metric on N k or Z k. The attentie reader will notice a potential conflict of notation between the p-adic metric and the pro-p metric on Z k, which were both denoted by d p. The net lemma gets rid of this ambiguity. 5

6 Lemma 1.2 The p-adic metric and the pro-p metric coincide on N k and on Z k. Proof. First, the pro-p metric on N k is the restriction to N k of the pro-p metric on Z k since eery morphism from N k onto a group etends uniquely to Z k. Therefore it suffices to proe that for all u, Z k, one has r p (u, = ν p (u. Since Z k is Abelian, a direct product of groups separates u and if and only if one of its factors separates u and. Let C r be the cyclic group of order r. In iew of the structure theorem of Abelian groups, one has r p (u, = min { n C p n+1 separates u and } ν p (u Further, for eery homomorphism f : Z k Z, u (mod p n+1 implies f(u f( (mod p n+1. Therefore u (mod p n+1 is a necessary condition for C p n+1 to separate u and. Thus r p (u, = ν p (u as claimed. There is a nice connection [11] between the pro-p metric and the binomial coefficients, which comes from the characterization of the languages recognized by a p-group gien by Eilenberg and Schützenberger [5, Theorem 10.1, p. 239]. Let us call a p-group language a language recognized by a p-group. Note that such a language is recognizable by definition. Proposition 1.3 A language of A is a p-group language if and only if it is a Boolean combination of the languages { } L(, r, p = u A r (mod p, for 0 r < p and A. Let us set now r p(u, = min { A and ( } (mod p d p(u, = p r p (u,. It is proed in [11, Theorem 4.4] that d p is an ultrametric uniformly equialent to d p. We shall use this fact in the proof of our main theorem. The following result, which follows from [15, Theorem 2.3] proides a characterization of uniform continuity in terms of p-group languages. Let A and B be two finite alphabets. Theorem 1.4 A function f : A B is uniformly continuous for d p if and only if, for eery p-group language L of B, the language f 1 (L is a p-group language of A. 6

7 Since p-group languages are closed under inerse of morphisms [5], one gets the following corollary. Corollary 1.5 Eery monoid morphism from A to B is uniformly continuous for d p. As another consequence of Theorem 1.4, we obtain: Corollary 1.6 For eery A, the function from A into N which maps eery word onto the binomial coefficient ( is uniformly continuous for d p. Proof. Let f : A N be the function defined by f ( = (. Since the only p-group quotients of N are the cyclic groups of order p n, a language of N recognized by a p-group is a finite union of arithmetic progressions r+p n N (r {0,..., p n 1}. In iew of Theorem 1.4, and since the class of p-group languages is closed under union, we only need to show that f 1 (r + p n N is a p-group language. Now, [10, Eample 4, p. 450] shows that f 1 (r + p n N = is indeed recognized by a p-group. { A ( } r (mod p n 2 Mahler s epansions The first step to etend Mahler s theorem to functions from words to integers is to define a suitable notion of Mahler s epansion for these functions. 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 inductiely an operator w for each word w A by setting ( 1 f(u = f(u, and for each letter a A, For instance, ( aw f(u = ( a ( w f(u. aab f(u = f(u + 2f(ua + f(ub f(uaa 2f(uab + f(uaab The net proposition gies a direct way to compute the difference operators. Proposition 2.1 For each word word w A, the following relation holds: w f(u = ( ( 1 w + w f(u (2.1 0 w 7

8 Proof. We proe the result by induction on the length of w. If w is the empty word, the result is triial. Let w be a word and let a be a letter. Then ( aw f(u = ( a ( w f(u = ( w f(ua ( w f(u and by the induction hypothesis applied to w, ( aw f(u = ( ( 1 w + w (f(ua f(u 0 w On the other hand, since ( aw {( w ( = az + w ( z w we get and thus 0 aw if = az otherwise ( ( 1 aw + aw f(u = ( 1 w + az +1(( w ( w + f(uaz az z 0 az w +1 ( + ( 1 w + +1 w f(u = 0 z w = 0 w ( ( 1 w + z w z ( aw f(u = 0 w +1 / aa f(uaz + ( ( 1 w + +1 w f(u 0 w ( ( 1 w + w (f(ua f(u 0 aw ( ( 1 aw + aw f(u which gies the induction step for the proof of (2.1. By Corollary 1.6, for a fied A, the function ( from A to N which maps a word u to the binomial coefficient ( u is uniformly continuous for dp. This family of functions, for ranging oer A, is locally finite in the sense that, for each u A, the binomial coefficient ( u is null for all but finitely many words. In particular, if (m A is a family of integers, there is a well-defined function from A to Z defined by the formula f(u = A m We can now state our first result, which doesn t require any assumption on f. 8

9 Theorem 2.2 Let f : A Z be an arbitrary function. Then there eists a unique family f, A of integers such that, for all u A, f(u = f, (2.2 A This family is gien by f, = ( f(1 = 0 0 ( ( 1 + f( (2.3 Proof. First obsere that, according to (2.1 ( f(1 = ( ( 1 + f( (2.4 Thus ( f(1 = A A = A ( 1 u + = f(u ( ( 1 + f( ( ( 1 + u u ( f( 0 u in iew of the following relation from [7, Corollary 6.3.8]: ( { ( 1 u + u ( 1 if u = w = w 0 otherwise 0 u Uniqueness of the coefficients f, follows inductiely from the formula f, u = f(u f,, a straightforward consequence of ( < u The series defined by (2.2 is called the Mahler s epansion of f. (2.5 Eample 2.1 Let A be the binary alphabet {0, 1}. The empty word will be eceptionally denoted by ε to aoid confusion. Let f : A N be the function mapping a binary word onto its alue as a binary number. For instance f( = f(10111 = 23. Then one sees easily that { ( f + 1 if 1{0, 1} f = f otherwise { ( 1 if 1{0, 1} f(ε = 0 otherwise 9

10 It follows that the Mahler epansion of f is f = ( 1{0,1} For instance, if u = 01001, one gets f(u = = = The Mahler s epansion of a function f : A Z can be used to compute its uniform norm f p, defined by f p = sup u A f(u p First obsere that f p = ma u A f(u p since f(u p takes alues in the set {p n n 0} {0}. Theorem 2.3 Let A f, ( be the Mahler s epansion of a function f : A Z. Then f p = ma A f, p. Proof. Let us choose a word u of A such that f p = f(u p. Then f(u = u f, and thus we get by (1.1 and (1.2 f p = f(u p ma u f, p ma A f, p Conersely, let w be a word of minimal length such that The formula f, w p = ma A f, p f, w = f(w 0 < w f, yields, with the help of (1.1 and (1.2, f, w p ma ( f(w p, ma f, p 0 < w Now, for 0 < w, one has f, p < f, w p by the minimality of w and thus ma 0 < w f, p < f, w p. It follows that f, w p f(w p f p and hence ma A f, p = f p as claimed. 10

11 3 Mahler polynomials A function f : A Z is a Mahler polynomial if its Mahler s epansion has finite support, that is, if the number of nonzero coefficients f, is finite. In this section, we proe in particular that Mahler polynomials are closed under addition and product. We first introduce a conenient combinatorial operation, the infiltration product. We follow the presentation of [7]. Let Z A be the ring of formal power series in noncommutatie indeterminates in A. Any series s is written as a formal sum s = u A s, u u, a notation not to be confused with our notation for the Mahler s epansion. The infiltration product is the binary operation on Z A, denoted by and defined inductiely as follows: for all u A, u 1 = 1 u = u, (3.1 for all u, A, for all a, b A { (u ba + (ua b + (u a if a = b ua b = (u ba + (ua b otherwise (3.2 for all s, t Z A, s t = u, A s, u t, (u (3.3 Intuitiely, the coefficient u, is the number of pairs of subsequences of which are respectiely equal to u and and whose union gies the whole sequence. For instance, ab ab = ab + 2aab + 2abb + 4aabb + 2abab ab ba = aba + bab + abab + 2abba + 2baab + baba Also note that u, u = ( u. We shall need the following relation (see [7, p.131]. For all 1, 2 A, ( ( u u = 1 2, ( A Formula 3.4 leads to an eplicit computation of the Mahler s epansion of the product of two functions. Proposition 3.1 Let f and g be two functions from A to N. Then the coefficients of the Mahler s epansion of fg are gien by the formula: fg, = f, 1 g, 2 1 2, 1, 2 A 11

12 Proof. Indeed, if f(u = A f, ( u and g(u = A g, ( u, then fg(u = ( ( u u f, 1 g, 2 1, 2 A 1 2 and the result follows by Formula (3.4. It is now easy to proe the result announced at the beginning of this section. Proposition 3.2 Mahler polynomials form a subring of the ring of all functions from A to Z for addition and multiplication. Proof. It is clear that the difference of two Mahler polynomials is a Mahler polynomial. Further Proposition 3.1 shows that Mahler polynomials are closed under product. 4 Uniform continuity In this section, we gie a simple characterization of the uniformly continuous functions from A to Z. Uniform continuity always refers to the metric d p, but it can be replaced by d p wheneer conenient in iew of [11, Theorem 4.4]. Theorem 4.1 Let A f, ( be the Mahler s epansion of a function f : A Z. Then f is uniformly continuous if and only if lim f, p = 0. Proof. Suppose that lim f, p = 0 and let r N. Then there eists s N such that, if s, ν p ( f, r. Setting g = ( f, <s and h = f, s ( we get f = g + h. Further p r diides f, for s. Since g is a Mahler polynomial, it is uniformly continuous by Corollary 1.6 and so there eists t N such that d p (u, u p t implies g(u g(u (mod p r and hence f(u f(u (mod p r. Thus f is uniformly continuous. This proes the easy direction of the theorem. The key argument for the opposite direction is the following approimation result. Theorem 4.2 Let f : A N be a uniformly continuous function. Then there eists a Mahler polynomial P such that, for all u A, f(u P (u (mod p. 12

13 Proof. We first proe the theorem for some characteristic functions related to the binomial coefficients. The precise role of these functions will appear in the course of the main proof. Let A and let s be an integer such that 0 s < p. Let χ s, : A N be the function defined by { 1 if χ s, (u = s (mod p 0 otherwise Lemma 4.3 There is a Mahler polynomial P s, such that, for all u A, χ s, (u P s, (u (mod p. Proof. Let P s, (u = [ ] [ ] [( 1 u ] (p 1 s Then P s, is a Mahler polynomial by Proposition 3.2. If s (mod p, then P s, (u 0 (mod p. If s (mod p, then by Ibn Al-Haytham s theorem (also known as Wilson s theorem, P s, (u (p 1! 1 (mod p It follows that P s, (u χ s, (u (mod p in all cases. We now proe Theorem 4.2. Since f is uniformly continuous for the metric d p, there eists a positie integer n such that if, for 0 n, then ( (mod p f(u f( (mod p It follows that the alue of f(u modulo p depends only on the residues modulo p of the family {( } u 0 n. Let R be the set of all families r = {r } 0 n such that 0 r < p. For 0 i < p, let R i be the set of all families r of R satisfying the following condition: if, for 0 n, r (mod p, then f(u i (mod p (4.1 The sets (R i 0 i<p are pairwise disjoint and their union is R. We claim that, for all u A, f(u ip i (u (mod p (4.2 0 i<p 13

14 where P i is the Mahler polynomial P i = r R i 0 n First consider, for r R, the characteristic function χ r (u = χ r,(u 0 n P r, (4.3 By construction, χ r is defined by { 1 if, for 0 n, χ r (u = r (mod p 0 otherwise and it follows from (4.1 and from the definition of R i that f(u ( i χ r (u (mod p (4.4 r R i 0 i<p Now Lemma 4.3 gies immediately χ r (u P r,(u (mod p (4.5 0 n and thus (4.2 follows now from (4.3, (4.4 and (4.5. The result follows, since P = ip i (u is a Mahler polynomial. 0 i<p Theorem 4.2 can be etended as follows. Corollary 4.4 Let f : A N be a uniformly continuous function. Then, for each positie integer r, there eists a Mahler polynomial P r such that, for all u A, f(u P r (u (mod p r. Proof. We proe the result by induction on r. For r = 1, the result follows from Theorem 4.2. If the result holds for r, there eists a Mahler polynomial P r such that, for all u A, f(u P r (u 0 (mod p r. Let g = f P r. Since g is uniformly continuous, there eists a positie integer n such that if ( (mod p for n, then g(u g( (mod p 2r. It follows that 1 p r g(u 1 p r g( (mod p r, and thus 1 p r g is uniformly continuous. Applying Theorem 4.2 to 1 p r g, we get a Mahler polynomial P such that, for all u A, 1 g(u P (u (mod p pr 14

15 Setting P r+1 = P r + p r P, we obtain finally which concludes the proof. f(u P r+1 (u (mod p r+1 We now come back to the proof of Theorem 4.1. For each positie integer r, there eists a Mahler polynomial P r such that, for all u A, f(u P r (u (mod p r. Using (2.3 to compute eplicitly the coefficients f P r,, we obtain f P r, 0 (mod p r Since P r is a polynomial, there eists an integer n r such that for all A such that n r, P r, = 0. It follows f, p p r and thus lim f, p = 0. 5 An etension of Mahler s theorem Mahler s theorem is often presented as an interpolation result (see for instance [9, p. 57]. This can also be etended to functions from words to integers. Gien a family of integers (c A, one can ask whether there is a (uniformly continuous function f from the free pro-p group to Z such that f( = c. The answer to this question is part of our main result: Theorem 5.1 Let A f, ( be the Mahler s epansion of a function f : A Z. Then the following conditions are equialent: (1 f is uniformly continuous for d p, (2 the sequence of partial sums 0 n f, ( conerges uniformly to f, (3 lim f, p = 0, (4 lim f p = 0, (5 f admits a continuous etension from F p (A to Z p. Proof. (1 implies (3 follows from Theorem 4.1. (3 implies (2. Formulas (1.1 and (1.2 gie for all u A and for all n N p f, sup f, p sup f, p n< u n< n< Let ε > 0. By (3, there is an integer N such that if N, then f, p < ε. It follows that for all n N and for all u A, n< f, p sup f, p < ε. n< 15

16 which proes (2, in iew of (2.1. (2 implies (1. Theorem 4.1 shows that eery Mahler polynomial is uniformly continuous. Now, if a sequence of uniformly continuous functions conerges uniformly, then its limit is uniformly continuous as well. Thus (2 implies that f is uniformly continuous. (3 implies (4. A straightforward induction shows that u ( f = u f holds for all u, A. Hence and Theorem 2.3 yields f, u = u ( f(1 = u f(1 = f, u f p = ma u A f, u p = ma u A f, u p (5.1 Let ε > 0. By (3, there eists some N N such that f, w p < ε wheneer w N. Thus, wheneer N, we get f, u p < ε and so f p < ε by (5.1. Therefore (4 holds. (4 implies (3 follows immediately from (5.1. (1 implies (5 since F p (A (respectiely Z p is the completion of A (respectiely Z for d p. (5 implies (1. Since F p (A is compact, the continuous etension of f to F p (A is uniformly continuous and its restriction f to A is uniformly continuous as well. 6 The commutatie case A Mahler epansion can also be obtained for functions from N k into Z. First obsere that the family of functions from N k into N {( ( ( } (r 1,..., r k N k r 1 r 2 is locally finite. Thus, gien a family (m r r N k, the formula f(n 1,..., n k = ( ( n1 nk r k r N k m r defines a function f : N k Z. Conersely, each function from N k to Z admits a unique Mahler epansion, a result proed in a more general setting in [2, 1]. Proposition 6.1 Let f : N k Z be an arbitrary function. Then there eists a unique family f, r r N k of integers such that, for all (n 1,..., n k N k, f(n 1,..., n k = ( ( n1 nk f, r r 1 r k r N k 16 r 1 r k

17 Furthermore, the coefficients are gien by f, r = r 1 i 1 =0... r k i k =0 ( 1 (r r k +(i i k ( r1 i 1 ( rk i k f(i 1,..., i k The net lemma relates commutatie and noncommutatie Mahler epansions. Gien a finite alphabet A, we denote by c the map from A to N A which maps eery word u onto its commutatie image c(u = ( u a a A. Lemma 6.2 Let ( ( r N k f, r be the Mahler epansion of a r 1 function f from N k into Z. Let A = {a 1,..., a k } be a k-letter alphabet. Then f, r = f c, a r 1 1 ar k k for eery r N k. Proof. One has f(n 1,..., n k = f(c(a n 1 1 a n k k = ( a n 1 1 a n k f c, k A = n 1 r 1 =0 n k r k... f c, a r 1 1 a r k r k =0 ( n1 k r 1 ( nk The uniqueness of the Mahler epansion stated in Proposition 6.1 yields the equality f, r = f c, a r ar k k for eery r N k. We can now deduce a new proof for the hard implication of the generalization of Mahler s Theorem to N k (see [2] or [18, Section 4.2.4]. Gien r N k, let us write r for r r k. Corollary 6.3 (Amice [2] Let ( ( r N k f, r be the Mahler r 1 epansion of a function f from N k into Z. Then f is uniformly continuous for d p if and only if lim r f, r p = 0. Proof. Assume that f is uniformly continuous. Let A = {a 1,..., a k }. Since c : A N k is a morphism, it is uniformly continuous by Corollary 1.5 and so is the function f c. By Theorem 4.1, we obtain lim f c, p = 0. In particular, lim r f c, a r 1 1 ar k k p = 0 and so lim r f, r p = 0 by Lemma 6.2. The opposite implication is the easy one (see ([2] and ( can be deduced from the fact that if n 1 n 2 (mod p s+νp(m! n1 n2 then (mod p s. m m Details are omitted. r k r k 17

18 7 Back to language theory Our original motiation was the study of regularity-presering functions f from A to B, in the following sense: if L is a regular language of B, then f 1 (L is a regular language of A. More generally, we are interested in functions presering a gien ariety of languages V: if L is a language of V, then f 1 (L is also a language of V. There is an important literature on the regular case [20, 6, 19, 12, 13, 14, 3]. A remarkable contribution to this problem can be found in [17], where a characterization of sequential functions presering star-free languages (respectiely group-languages is gien. A similar problem was also recently considered for formal power series [4]. An open problem is to characterize the functions from A to B presering the p-group languages. Theorems 1.4 and 5.1 gie a solution when B is a one-letter alphabet. Theorem 7.1 Let f be a function from A to N. Then the following conditions are equialent: (1 for each p-group language L of N, f 1 (L is a p-group language of A, (2 f is uniformly continuous for d p, (3 lim f p = 0. It would be interesting to find a suitable etension of this result for functions from A to B, for any alphabet B. Another challenge would be to etend other results from p-adic analysis to the word case. Acknowledgement The authors would like to thank also Daniel Barsky and Gilles Christol for pointing out seeral references. References [1] R. Ahlswede and R. Bojanic, Approimation of continuous functions in p-adic analysis, J. Approimation Theory 15,3 (1975, [2] Y. Amice, Interpolation p-adique, Bull. Soc. Math. France 92 (1964, [3] J. Berstel, L. Boasson, O. Carton, B. Petazzoni and J.-É. Pin, Operations presering recognizable languages, Theoret. Comput. Sci. 354 (2006, [4] M. Droste and G.-Q. Zhang, On transformations of formal power series, Inform. and Comput. 184,2 (2003,

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20 [18] A. M. Robert, A course in p-adic analysis, Graduate Tets in Mathematics ol. 198, Springer-Verlag, New York, [19] J. I. Seiferas and R. McNaughton, Regularity-presering relations, Theoret. Comp. Sci. 2 (1976, [20] R. E. Stearns and J. Hartmanis, Regularity presering modifications of regular epressions, Information and Control 6 (1963,