ERGODICITY OF p-ADIC MULTIPLICATIONS AND THE DISTRIBUTION OF FIBONACCI NUMBERS

We characterise ergodicity of p-adic multiplication on the units of Zp where p is an odd prime. We do the same for the subgroup of units of Zp( √ D) of norm ± 1, when D is not a square in Zp. These results are then applied to give a complete description of the distribution mod p of the Fibonacci numbers. Introduction The main direction of research into the distribution mod p of linear recurrence sequences seems to lean toward problems associated with uniform distribution (see for instance [Nie, KS, KN, Bum, Nat, Nar, Tur, Ve]). Enquiries into other distributions have not, so far, received the same attention. The purpose of this paper is to consider this more general question in the concrete setting of the Fibonacci sequence. (In this case one has uniform distribution mod p only when p = 5.) By restricting ourselves to this special case we are able to give a complete characterisation of the asymptotic distribution mod p, for all k > 0, where p is an odd prime. However, we feel confident that our methods, involving the elementary ergodic theory of p-adic multiplication, will prove to be of use to more general linear recurrence sequences and we expect to return to this question elsewhere. For an odd prime p we consider the multiplicative group U of units of the ring Zp of p-adic integers. For each λ ∈ U , the map Tλ : U → U given by Tλ(x) = λx is the restriction of an automorphism of Zp and as such preserves Haar measure. The following characterises the ergodicity of Tλ. Theorem 1. Let 1 6= λ ∈ U then Tλ : U → U is ergodic if and only if λ is a primitive root mod p and λ 6≡ 1 (mod p). Moreover, all such transformations are isomorphic. This is an immediate consequence of the following result which gives a complete description of the ergodic decompositions of all maps Tλ, λ 6= 1. Theorem 2. If λ has order m mod p (m | p − 1) and λ = 1 + μp where μ is a unit then Tλ has p (p − 1)/m ergodic components. The restriction of Tλ to these components are all isomorphic to the map of (Z/mZ)×Zp defined by (i, x) 7→ (i+ 1 mod m, x+ 1). 1991 Mathematics Subject Classification. Primary 11B39, 28D05; Secondary 11N69, 11K41. The first named author acknowledges partial support from the European Science Foundation Programme PRODYN during part of this work. 1 2 ZAQUEU COELHO AND WILLIAM PARRY We take Zp to be the ring of sums x = ∑∞ n=0 xnp n (xn = 0, 1, · · · , p − 1) whose convergence is assured with respect to the p-adic metric. This metric gives the infinite direct product topology on {0, 1, · · · , p−1} + which is compact and zero dimensional. (An element is close to zero when it is divisible by p forN large.) The units (invertible elements) of Zp consist of those x ∈ Zp with x0 6= 0, i.e. U = {x : p 6 | x}. Since one can write Zp = ∪n=0 pU ∪ {0}, once we have the ergodic decomposition structure of Tλ : U → U we have it for Tλ : Zp → Zp. (The structure is copied on each pU .) If D ∈ U is not a square Zp( √ D) = Zp+ √ DZp is a ring with the obvious addition, multiplication and topology. (Again x+ √ Dy is close to zero if both x, y are divisible by a high power of p.) The units U( √ D) consist of all x + √ Dy with one of x, y a unit of Zp. Since the expression x + √ Dy is unique we can define the norm N by N(x+ √ Dy) = x2−Dy2 and this is a homomorphism of U( √ D) into U with ‘kernel’ U ( √ D) = { x+ √ Dy : x −Dy = ± 1 } . Along with the maps Tλ : U → U we shall consider the maps Tλ : U( √ D) → U ( √ D) where λ ∈ U ( √ D). The cardinality of U ( √ D) mod p is clearly p2−1, whereas the order of U( √ D) mod p is 2(p+ 1). To see this we note that each element of Fp is the mod p norm of some element of U ( √ D) and it is not difficult to see there is a one-one correspondence between {x ∈ U ( √ D) mod p : N(x) = k} and {x ∈ U( √ D) : N(x) = 1}. For this reason each of these sets has cardinality (p2−1)/(p−1) = p+1, and therefore U ( √ D) has order 2(p+1). Furthermore, as U( √ D) mod p is a subgroup of the multiplicative group of a finite field we see that U( √ D) is cyclic. The analogue of Theorem 1 for U ( √ D) is Theorem 3. For λ ∈ U( √ D) the map Tλ : x 7→ λx is ergodic on U ( √ D) if and only if λ is primitive modp (i.e. 2(p + 1) is the least positive integer k for which λ = 1 mod p) and λ 6≡ 1 (mod p). There is a corresponding analogue of Theorem 2 in the sense that there are finitely many Tλ ergodic components for arbitrary λ ∈ U( √ D). These components can be completely described in terms of the order m of λ (which must divide 2(p + 1)) and the largest k for which λ = 1 mod p. Since the proof follows the same lines as Theorems 2 and 3 we shall omit a precise statement and demonstration. In Section 3 we specialise to the case D = 5 in order to study the Fibonacci sequence mod p where p 6= 2, 5 is prime. (But see our remarks for the case p = 5.) Defining N(z) = 1 or −1 according to whether z is a square or not (in U or U ( √ 5)) we let f be the function f : U → Zp (U( √ 5) → Zp) given by f(z) = 1 √ 5 (