Metric properties of some special p -adic series expansions

LXXVI.1 (1996)

Metric properties of some special p -adic series expansions

by

Arnold Knopfmacher and John Knopfmacher (Johannesburg)

1. Introduction. Let Q be the field of rational numbers, p a prime number and Q

the completion of Q with respect to the p-adic valuation | |

defined on Q by

(1.1) |0|

= 0 and |A|

= p

if A = p

r/s, where p - r, p - s.

Then Q

is the field of p-adic numbers with p-adic valuation | |

, the exten- sion of the original valuation on Q (cf. Koblitz [12] or Schikhof [19]).

It is well known that every A ∈ Q

has a unique series representation A = P

n=v(A)

c

p

, c

∈ {0, 1, 2, . . . , p − 1}. In the discussion below we call the finite series hAi = P

v(A)≤n≤0

c

p

the fractional part of A. Then hAi ∈ S

, where we define S

= {hAi : A ∈ Q

} ⊂ Q.

This set S

is not multiplicatively or additively closed. The function hAi and set S

have been used in the study of certain types of p-adic continued fractions by Mahler [14], Ruban [17] and Laohakosol [13] in particular.

Recently the fractional part hAi was used by the present authors [7], [8]

to derive some new unique series expansions for any element A ∈ Q

, in- cluding in particular analogues of certain “Sylvester”, “Engel” and “L¨ uroth”

expansions of arbitrary real numbers into series with rational terms (cf. [16], Chap. IV). It turns out that p-adic and real L¨ uroth series may be regarded in some sense as algorithmic relatives of continued fractions, and there is some interest in studying possible parallels between the algorithms or digits inducing them. In the direction of metric and asymptotic results concerning digits, various analogies of this kind were previously established, especially by Jager and de Vroedt [5] and Sal´at [18] for real L¨ uroth series, and by Ruban [17] for p-adic continued fractions, in comparison with classical the- orems of Khinchin (see e.g. [2], [6]) for real continued fractions. (In these developments, Haar measure for p-adic numbers replaces Lebesgue measure for real numbers.)

The main aim of the present paper is to state or derive some similar metric and asymptotic results for the p-adic L¨ uroth-type expansions re-

[11]

ferred to above. Here, as in other areas such as e.g. transcendence and diophantine approximation theory (cf. Sprindˇzuk [20]), there are also par- allels with results in the partly analogous but different context of Lau- rent series over finite fields; see Paysant-Le Roux and Dubois [15] and the present authors [9], [10]. Consequently, many of the steps below are given only in outline, together with references to fuller parallel arguments where appropriate. (We thank an anonymous referee for some helpful com- ments.)

2. L¨ uroth-type algorithm and ergodic properties. The L¨uroth-type expansion (see (2.1) below) of a p-adic number A ∈ Q

was derived in [7]

from the following algorithm for the “digits” a

= a

(A) ∈ S

: Define a

= hAi and A

= A − a

and observe that

a

= c ∈ S

⇔ v(A − c) ≥ 1 ⇔ A − c ∈ X

,

where X

= pZ

is the maximal ideal in the ring Z

of all p-adic integers, i.e. p-adic numbers of order ≥ 0. If A

6= 0 (n ≥ 1) has already been defined, inductively define

a

= h1/A

i and A

= (a

− 1)(a

A

− 1),

so that v(a

) ≤ −1 for n ≥ 1. If any A

= 0, or a

= 0, stop the algorithm.

This algorithm leads (cf. [7]) to a finite or convergent (relative to | |

) expansion

(2.1) A = a

+ 1

a

+ X

n≥2

1

a

(a

− 1) . . . a

(a

− 1)a

,

which is unique for A subject to the stated condition on the digits a

. Another way of looking at it is in terms of operators a and T (where a : X

\ {0} → S

, T : X

→ X

) such that a(x) = h1/xi, T (0) = 0 and otherwise T (x) = (a(x) − 1)(xa(x) − 1). Then for x = A

∈ X

we have a

= a

(x) = a(x) and more generally a

= a

(x) = a

(T

x) if 0 6=

T

x ∈ X

.

Although the conclusions of the next theorem are sharpened in Section 3 below it seems at least worth sketching briefly how they can also be deduced from the Ergodic Theorem, after proving that x ∈ X

⇒ T (x) ∈ X

and that the resulting operator T : X

→ X

is ergodic relative to Haar measure µ on X

.

Theorem 1. (i) For any given k ∈ S

with v(k) ≤ −1, and all x ∈ X

outside a set of Haar measure 0, the digit value k has asymptotic frequency

n→∞

lim 1

n #{r ≤ n : a

(x) = k} = |k|

.

(ii) For all x ∈ X

outside a set of Haar measure 0 there exists a single

“Khinchin-type” constant

n→∞

lim |a

(x) . . . a

(x)|

= p

. (iii) For all x ∈ X

outside a set of Haar measure 0,

|x − w

|

= p

as n → ∞, where

w

= w

(x) = X

λ

a

, λ

= 1, λ

= 1

a

(a

− 1) . . . a

(a

− 1) . For this and later theorems, a convenient description of the Haar measure µ on X

= pZ

is given in Sprindˇzuk [20], pp. 67–70. In particular, µ(C) = p

for any “circle”, “disc” or “ball”

C = C(x, p

) := {y ∈ Q

: |y − x|

≤ p

} of radius p

. So µ(X

) = 1, since X

= C(0, p

).

Now note that every “digit” a(x) lies in the set S

:= {hAi : v(A) ≤ −1}.

For any given digits k

, . . . , k

∈ S

, let

I

= I

(k

, . . . , k

) := {x ∈ X

: a

(x) = k

, . . . , a

(x) = k

} and call I

a basic (L¨uroth) cylinder of rank n. Also let I

= X

.

The L¨ uroth-type expansion (2.1) of any x ∈ I

then has the form x = w

+ λ

X

r>n

1

a

(a

− 1) . . . a

(a

− 1)a

, where

λ

= 1, λ

= 1

k

(k

− 1) . . . k

(k

− 1) for 1 ≤ r ≤ n, and

w

= X

λ

k

.

Thus x = w

+ λ

T

(x) = ψ

(T

(x)), if ψ

= ψ

(k

, . . . , k

) : X

→ I

is defined by ψ

(y) = w

+ λ

y (y ∈ X

). The “linear-type” map ψ

is then 1-1 onto, with inverse map T

: I

→ X

. In particular, I

= Im(ψ

) = w

+λ

X

. Since X

= C(0, p

), it then follows that I

= C(w

, p

|λ

|

) and has Haar measure µ(I

) = p

= |λ

|

. Hence

(2.2) µ(I

) = 1

|k

(k

− 1) . . . k

(k

− 1)|

= 1

|k

. . . k

|

,

since v(k) = v(k − 1) for v(k) ≤ −1.

More generally, for any r

< . . . < r

, we obtain

(2.3) µ{x ∈ X

: a

(x) = k

, . . . , a

(x) = k

} = |k

. . . k

|

. In particular, µ{x ∈ X

: a

(x) = k} = |k|

for any r ≥ 1 and k ∈ S

. Thus the digit functions are identically distributed and independent random variables relative to µ.

Now, in a standard way quite similar to that followed by Jager and de Vroedt [5] for real L¨ uroth series, one can deduce that T is measure- preserving and ergodic. (In fact, the stronger Bernoulli property for T could be approached along lines analogous to some given in [1].)

Theorem 1 and some further conclusions then follow by making special choices for integrable functions f in the ergodic formula

(2.4) lim

n→∞

1 n

X

f (T

x) = \

Xp

f dµ a.e.

For example, part (i) of Theorem 1 follows from consideration of the char- acteristic function f

of a basic cylinder I

(k). Alternatively, use of the function b f (·) = log

|a

(·)|

leads to the limit

(2.5) lim

n→∞

1 n

X

log

|a

(x)|

= \

Xp\{0}

f dµ = b p

p − 1 a.e.,

and this implies part (ii) of Theorem 1. The same function b f may be used in the deduction of part (iii), in combination with the following inequalities analogous to some appearing for Laurent series in [10]:

(2.6) 1 −

n+1

X

r=1

log

|a

(x)|

≤ log

|x − w

|

≤ −1 − X

log

|a

(x)|

. The function b f may also be used to show that the operator T has entropy

(2.7) h(T ) = − lim

n→∞

1

n log

µ(I

) = 2p log

p p − 1 .

Lastly, it is interesting to note that, in contrast to (2.5), a truncation argument involving the function e f (·) = |a

(·)|

leads to the conclusion

(2.8) lim

n→∞

1 n

X

|a

(x)|

= ∞ a.e.

3. Sharper asymptotic estimates. The fact that the p-adic L¨ uroth-

type digit functions a

(·) define identically distributed and independent ran-

dom variables on X

paves the way for the introduction of methods and

results of probability theory, which lead to sharper and deeper results than those considered earlier.

In the first place, the law of the iterated logarithm and the central limit theorem (cf. Theorems 3.16/17 in Galambos [4]) yield:

Theorem 2. Let A

(x) = #{r ≤ n : a

(x) = k}. Then for almost all x ∈ X

,

lim sup

n→∞

A

(x) − n|k|

√ n log log n = q

2|k|

(1 − |k|

) . Further , for any real z,

n→∞

lim µ



x ∈ X

: A

(x) − n|k|

< z

|k|

q

n(1 − |k|

)



= 1

√ 2π

z

\

−∞

e

du.

Now define a sequence (t

) of independent random variables t

on X

by t

(x) =

 v(a

(x)) if |a

(x)|

≤ n

,

0 otherwise.

Then the expected value E(t

) = X

p^r≤n²

p

r(p − 1)p

= E(v(a

(·))) + O

 log n n

 ,

and a similar calculation of E(t

) and the variance var(t

) leads to the estimate

(3.1) B

:=

X

var(t

) = pn

(p − 1)

+ O(1) as n → ∞.

Next, since t

(x) ≤ 2 log

n = o( p

B

/ log log B

), the law of the iterated logarithm implies

(3.2) lim sup

n→∞

P

r=1

t

− P

r=1

E(t

)

√ 2B log log B

= 1 a.e.

Hence

(3.3) lim sup

n→∞

P

r=1

t

− P

r=1

E(v(a

(·))) q

2

n log log n = 1 a.e.

Now let U

= {x ∈ X

: t

(x) 6= v(a

(x))}. Then µ(U

) = X

|k|_p>n²

|k|

< 1

n

,

and the Borel–Cantelli lemma yields µ(lim sup

U

) = 0. Thus, for al- most all x ∈ X

, there exists n

(x) with t

(x) = v(a

(x)) for n ≥ n

(x).

Therefore (3.3) now implies:

Theorem 3. For almost all x ∈ X

, lim sup

n→∞

P

r=1

v(a

(x)) − c

n

√ n log log n = √ 2c

, where c

= p/(p − 1), c

= p/(p − 1)

. Hence as n → ∞,

|a

(x) . . . a

(x)|

= p

+ O

r log log n n

 a.e.

The next theorem sharpens the last part of Theorem 1 above:

Theorem 4. If w

= w

(x) is defined as in Theorem 1(iii) then lim sup

n→∞

v(x − w

) + 2pn/(p − 1)

√ n log log n =

√ 8p

p − 1 a.e.

Hence

1

n v(x − w

) = − 2p p − 1 + O

r log log n n

 a.e.

By symmetry as in Feller [3], p. 205, Theorem 3 leads to (3.4) lim inf

n→∞

P

r=1

v(a

(x)) − 2pn/(p − 1)

√ n log log n = −2 √

2c

a.e.

In combination with (2.6) above, this implies Theorem 4.

4. Average and individual estimates for digits. By (2.8) above, the average

1 n

X

|a

(x)|

→ ∞ a.e. on X

as n → ∞. Theorem 5 estimates this average in probability over X

: Theorem 5. For any fixed ε > 0,

n→∞

lim µ



x ∈ X

: 1 n log

n

X

|a

(x)|

− (p − 1)    > ε



= 0, i.e.

1 n log

n

X

|a

(x)|

→ p − 1 in probability over X

.

P r o o f. Consider the truncation method of Feller [3], Chapter 10, §2, as applied to the random variables U

, V

(r ≤ n) defined by

U

(x) = |a

(x)|

, V

(x) = 0 if |a

(x)|

≤ n log

n, U

(x) = 0, V

(x) = |a

(x)|

if |a

(x)|

> n log

n.

In that case (4.1) µ



x ∈ X

: 1 n log

n

X

|a

(x)|

− (p − 1)    > ε



≤ µ{x : |U

+ . . . + U

− (p − 1)n log

n| > εn log

n}

+ µ{x : V

+ . . . + V

6= 0}, and

µ{x : V

+ . . . + V

6= 0} ≤ nµ{x : |a

(x)|

> n log

n}

(4.2)

= n X

|k|_p>n log_pn

|k|

< 1/ log

n = o(1).

Next E(U

+ . . . + U

) = nE(U

) and var(U

+ . . . + U

) = n var(U

), where E(U

) = X

|k|_p≤n log_pn

|k|

= (p − 1) log

([n log

n]) (4.3)

∼ (p − 1) log

n as n → ∞ and

(4.4) var(U

) < E(U

) = X

|k|_p≤log_pn

1 < pn(log

n).

Theorem 5 then follows from an application of (4.3) and (4.4) to Chebyshev’s inequality:

(4.5) µ{x : |U

+ . . . + U

− nE(U

)| > εnE(U

)} ≤ n var(U

)

(εnE(U

))

= o(1).

Note that the conclusion of Theorem 5 is not valid with probability one, since Theorem 3.13 in Galambos [4] implies that either

lim sup

n→∞

1 n log

n

X

|a

(x)|

= ∞ a.e.

or

lim inf

n→∞

1 n log

n

X |a

(x)|

= 0 a.e.

Regarding estimates for individual digits, now consider:

Theorem 6. Given any positive increasing function ψ(n) of n,

|a

(x)|

= O(ψ(n)) a.e. ⇔ X

1/ψ(n) < ∞.

In fact, |a

(x)|

= O(ψ(n)) is false a.e. if the series diverges.

P r o o f. Let V

= {x ∈ X

: |a

(x)|

> ψ(n)}. Since µ{x : a

(x) = k}

= |k|

by (2.3), it follows that µ(V

) = X

|k|p>ψ(n)

|k|

≤ p/ψ(n).

If P

n=1

ψ(n)

< ∞, then the Borel–Cantelli lemma (cf. [4], p. 36) yields µ(lim sup V

) = 0. Hence |a

(x)|

> ψ(n) for at most finitely many n, for almost all x ∈ X

. Thus |a

(x)|

= O(ψ(n)) a.e.

If P

n=1

ψ(n)

diverges, the Abel–Dini theorem (Knopp [11], p. 290) implies that there exists a positive increasing function θ(n) with θ(n) → ∞ as n → ∞ such that P

n=1

ψ(n)

θ(n)

also diverges. Then let W

= {x ∈ X

: |a

(x)|

> ψ(n)θ(n)}. The independence of the random variables a

implies the independence of the sets W

. Also

X

µ(W

) = X

X

|k|p>ψ(n)θ(n)

|k|

> 1 p

X

1

ψ(n)θ(n) = ∞.

Thus the Borel–Cantelli lemma yields µ(lim sup W

) = 1, and so |a

(x)|

>

ψ(n)θ(n) holds with probability one, for infinitely many n. Thus |a

(x)|

= O(ψ(n)) is false a.e.

Corollary. For almost all x ∈ X

, lim sup

n→∞

log |a

(x)|

− log n log log n = 1.

P r o o f. Theorem 6 implies that |a

(x)|

= O(n(log n)

) a.e. for any α > 1, while |a

(x)|

= O(n(log n)

) is false a.e. for any β ≤ 1. The corollary then follows by choosing α = 1 + ε, β = 1 − ε (ε > 0).

(Note that the corresponding lower limit is not finite a.e., since (2.3) earlier shows that |a

(x)|

can take any particular constant value p

(N ≥ 1) for all n, and all x in a set of positive measure.)

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A. Knopfmacher J. Knopfmacher

Department of Computational & Department of Mathematics

Applied Mathematics University of the Witwatersrand

University of the Witwatersrand Johannesburg, 2050, South Africa Johannesburg, 2050, South Africa

E-mail: arnoldk@gauss.cam.wits.ac.za

Received on 1.12.1994

and in revised form on 18.7.1995 (2703)