Representation of p-adic numbers in rational base numeration systems

Transcription

1 Representation of p-adic numbers in rational base numeration systems Christiane Frougny LIAFA, CNRS, and Université Paris 8 Numeration June 4-8, 200, Leiden Joint work with Karel Klouda

2 Base p = 0 Introduction to p-adic numbers 3 =

3 Base p = 0 What is the meaning of Introduction to p-adic numbers 3 = = 0 3

4 Base p = 0 What is the meaning of Introduction to p-adic numbers 3 = = = 7+6 i (0)i using the formal sum i 0 X i = X

5 Base p = 0 What is the meaning of Introduction to p-adic numbers 3 = = = 7+6 i (0)i using the formal sum i 0 X i = X Arithmetic as usual =

6 p-adic numbers The set of p-adic integers is Z p = i 0 a ip i with a i in A p = {0,,..., p }. Notation: a 2 a a 0. Z p. The set of p-adic numbers is Q p = i k 0 a i p i with a i in A p. Notation: a 2 a a 0.a a k0 Q p. Q p is a commutative ring, and Z p is a subring of Q p. p-adic valuation v p : Z\{0} Z is given by For x = a b Q n = p vp(n) n with p n. vp(x) a x = p b with (a, b ) = and (b, p) =. The p-metric is defined on Q by d p (x, y) = p vp(x y). It is an ultrametric distance.

7 Example p = 0 n {}}{ Take z n = Then z n 3 = 2 3 (0)n+ and v 0 (z n 3 ) = n+ thus d 0 (z n, 3 ) = (0) n 0 Hence the 0-adic representation of 3 is

8 p = 0 Take s = and t = Hence s + t = = s = t = s + t

9 = s = t = s t Hence st = 0 0 thus Z 0 is not a domain.

10 = s = t = s t Hence st = 0 0 thus Z 0 is not a domain. One has also s 2 = s, t 2 = t.

11 = s = t = s t Hence st = 0 0 thus Z 0 is not a domain. One has also s 2 = s, t 2 = t. One can prove that s = 2 0 and s = 5 t = 2 and t = 5 0.

12 Computer arithmetic 2 s complement In base 2, on n positions. n = 8: 0 = = = 0 Troncation of 2-adic numbers. = = 28

13 p prime Q p is a field if and only if p is prime. Q p is the completion of Q with respect to d p. p-adic absolute value on Q: { 0 if x = 0, x p = p vp(x) otherwise. Q p is the uotient field of the ring of p-adic integers Z p = {x Q p x p }.

14 p prime p-expansion of p-adic numbers Algorithm Let x = s t, s an integer and t a positive integer. (i) If s = 0, return the empty word a = ε. (ii) If t is co-prime to p, put s 0 = s and for all i N define s i+ and a i A p by Return a = a 2 a a 0. s i t = ps i+ t + a i. (iii) If t is not co-prime to p, multiply s t by p until xp l is of the form s t, where t is co-prime to p. Then apply (ii) returning a = a 2 a a 0. Return a = a a 0 a a 2 a l = a l a l a a 0. a is said to be the p-expansion of x and denoted by < x > p. l = v p (x)

15 Theorem Let x Q p. Then the p-representation of x is. uniuely given, 2. finite if, and only if, x N, 3. eventually periodic if, and only if, x Q.

16 Rational base numeration system (Akiyama, Frougny, Sakarovitch 2008) p > co-prime integers.

17 Rational base numeration system (Akiyama, Frougny, Sakarovitch 2008) p > co-prime integers. Representation of the natural integers: Algorithm (MD algorithm) Let s be a positive integer. Put s 0 = s and for all i N: Return s i = ps i+ + a i with a i A p p -expansion of s: < s > p s = n i=0 a i = a n a a 0. ( ) p i

18 p -expansions properties L p L p = {w A p w is is prefix-closed, any u A + p is a suffix of some w L L p is not context-free (if ), p -expansion of some s N} π : A + p Q the evaluation map. If v, w L p, then p, v w π(v) π(w).

19 T p p = 3, = 2 tree of nonnegative integers G 0 G G 2 G 3 G 4 Children of the vertex n are given by (pn+a) N, a A p. p G 0 =, G i+ = i G

20 In AFS 2008, the right infinite words that label the infinite paths of T p are defined as the admissible p -expansions of positive real numbers, of the form.a a 2, and it is proved that every positive real in [0, ] has exactly one p -expansion, but for an infinite countable subset of reals which have more than one such expansion. No periodic expansion. Connection with the problem of the distribution of the powers of a rational number modulo (Mahler 968).

21 MD algorithm the negative case Let s be a negative integer. The -expansion of s is < s > p= a 2 a a 0. from the MD algorithm: s 0 = s, s i = ps i+ + a i. p

22 MD algorithm the negative case Let s be a negative integer. The -expansion of s is < s > p= a 2 a a 0. from the MD algorithm: Properties of (s i ) i 0 : (i) (s i ) i is negative, s 0 = s, s i = ps i+ + a i. (ii) if s i < p p, then s i < s i+, (iii) if p p s i < 0, then p p s i+ < 0. p p p 0

23 In which fields does it work? We want: n s i=0 a i ( p n n s = s n + ) i=0 a i ( ) p n ( ) p n ( p n = s n r 0 ) r r as n Hence, if p = r j r j 2 2 r j k k, the p -expansion of s works only in Q rl,l =,...,k. The speed of convergence is then r jln.

24 p -expansions of negative integers Proposition Let k be a positive integer, and denote B = p p. Then: (i) if k B, then < k > p= ω b with b = k(p ), (ii) otherwise, < k > p= ω bw with w A + p and b = B(p ).

25 p = 3, = 2 T p tree of negative integers ω 0 ω T p ω T p

26 p = 3, = 2 T p ω 0 tree of negative integers ω - ω F 0 F F F F p p F 0 = B =, F i+ = i p F 2 0 F

27 p = 8, = 5 Trees T p and T p ω 0 ω 3 p = 8, = ω

28 p = 8, = 5 Trees T p and T p ω 0 ω p = 8, = ω The trees T p and T p are isomorphic if and only if p p Z.

29 p -expansions of negative integers properties for = and p prime we get the standard p-adic representation, L p = {w A p ω bw is p -expansion of s B, b / Pref(w)} L p is prefix-closed, any u A + p is a suffix of some w L L p is not context-free (if ). p,

30 MD algorithm for rationals Algorithm (MD algorithm) Let x = s t, s, t Z co-prime, s 0, and t > 0 co-prime to p. Put s 0 = s and for all i N define s i+ and a i A p by Return the s i t = p s i+ t + a i. p -expansion of x: < x > p= a 2 a a 0. s = i=0 a i ( ) p i

31 Examples x < x > p (s i ) i 0 ) abs. values p = 3, = , 3, 2,, 0, 0,... all -5 ω 202-5,-3,-2,-2,-2,... 3 /4 20,6,4,0,0,... all /8 ω 222,2,-4,-8,-8,-8,... 3 /5 ω (02)22,4,,-,-4,-6,-4,-6,... 3 p = 30, = ,, 0, 0,... all -5 ω ,-2,-,-,... 2, 3, 5 /7 ω (2 2 5) 23 3,, -5, -3, -6, -5,... 2, 3, 5

32 MD algorithm properties For the seuence (s i ) i from the MD algorithm we have: (i) if s > 0 and t =, (s i ) i is eventually zero, (ii) if s > 0 and t >, (s i ) i is either eventually zero or eventually negative, (iii) if s < 0, (s i ) i is negative, (iv) if s i < p p t, then s i < s i+, (v) if p p t s i < 0, then p p t s i+ < 0. p p t - 0

33 MD algorithm properties For the seuence (s i ) i from the MD algorithm we have: (i) if s > 0 and t =, (s i ) i is eventually zero, (ii) if s > 0 and t >, (s i ) i is either eventually zero or eventually negative, (iii) if s < 0, (s i ) i is negative, (iv) if s i < p p t, then s i < s i+, (v) if p p t s i < 0, then p p t s i+ < 0. Proposition Let x = s t Q. Then < x > p less than p p t. is eventually periodic with period

34 p -representation of r-adic numbers r, r, r 2,... prime numbers. p an integer >. Definition A left infinite word a k0 +a k0, k 0 N, over A p is a p -representation of x Q r if a k0 > 0 or k 0 = 0 and with respect to r. a i x = i= k 0 ( ) p i

35 Let p = r j r j 2 2 r j k k. Theorem Let x Q ri for some i {,..., k}. (i) If k = (i.e. p is a power of a prime), there exists a uniue p -representation of x in Q r. (ii) If k >, there exist uncountably many p -representations of x in Q ri.

36 Let p = r j r j 2 2 r j k k. Theorem Let x Q ri for some i {,..., k}. (i) If k = (i.e. p is a power of a prime), there exists a uniue p -representation of x in Q r. (ii) If k >, there exist uncountably many p -representations of x in Q ri. Let x Q and let i,..., i l {,...,k},l < k, be distinct. (i) There exist uncountably many p -representations which work in all fields Q ri,...,q ril. (ii) There exists a uniue p fields Q r,...,q rk ; namely, the -representation which works in all p -expansion < x > p. (iii) The p -expansion < x > p is the only p -representation which is eventually periodic.

37 p = 30, = The following are aperiodic p -representations of in both fields Q 2 and Q 3 :

38 p-odometer a a 0,...,p 2 p p 0

39 p-odometer a a 0,...,p 2 p p 0 p -odometer a a 0,..., p p p 0,...,p

40 Conversion from p-expansions to p -expansions Right on-line denumerable transducer: z 0 = 0, i = 0 and, for i 0 with a, b A p such that (z i, i) a b (z i+, i + ), a i + z i = b + p z i+. Input: a a 0 N A p such that x = i=0 a ip i Z r with r prime factor of p Output: b b 0 N A p such that x = b i i=0 The converter cannot be finite. ( p ) i.

41 Base 3 to system convertor 0, 0 0 0, ,, , 2, 2 2, 2 3, 2

42 On N A p define a distance δ: for a, b N A p, δ(a, b) = 2 i with i = min{j N a j b j }. Proposition The conversion from p-expansions to p -expansions χ : N A p a N A p b such that x = i=0 a ip i = ( ) i b i p i=0 Z r realized by the converter is Lipchitz, and thus uniformly continuous for the δ-topology. Remark: the inverse conversion is also realizable by a right on-line transducer. Corollary The digits in a p -expansion are uniformly distributed.

43 p -representations Algorithm (MDbis algorithm) Let x = s t, s, t Z co-prime, s 0, and t > 0 co-prime to p. Put s 0 = s and for all i N define s i+ and a i A p by s i t = p s i+ t + a i. Return the p -expansion of x: < x > p = a 2 a a 0. s = ( p a i i=0 ) i

44 All the results on finiteness and periodicity for p -representations are similar for p -representations. The main difference is on the tree of representations of the integers.

45 All the results on finiteness and periodicity for p -representations are similar for p -representations. The main difference is on the tree of representations of the integers. Theorem There exists a finite right seuential transducer C converting the p -representation of any x Z r, r prime factor of p, to its p -representation; the inverse of C is also a finite right seuential transducer. There is a one-to-one mapping between the sets of all p - and p -representations of elements of Z r which preserves eventual periodicity. One can say that the p - and p - numeration systems are isomorphic.

46 Right seuential transducer from p to p p = 3, = 2 0, ,

47 Negative rational base One can also define representations of the form ai ( p ) i, and ( ai p ) i with a i A p, by modifying the MD and MDbis algorithms by and s i t s i t = p s i+ t = p s i+ t + a i + a i

48 Negative rational base Definition (Kátai and Szabó 975) A Canonical Number System is a positional numeration system in which every integer (positive or negative) has a uniue finite expansion of the form a n a 0. Proposition The ( p )- and the ( p )- numeration systems are CNS. The ( p )- numeration system has been previously considered by Gilbert (99).

49 Conversions Theorem The conversion from the p (resp. the p ) -representation of any x Z r, r prime factor of p, to its ( p ) (resp. its ( p )) representation is realizable by a finite right seuential transducer ; the inverse conversion as well. p -rep. rational base ( ) p -rep. integer base p-rep. p -rep. ( ) p -rep. ( p)-rep. finite conversion by finite seuential transducer infinite conversion by on-line algorithm