Continued Fractions: arithmetic and ergodicproperties

Karma Dajani

May 6, 2015

2

1.1 Introduction and Basic Properties

It is well-known for centuries that any real number x ∈ [0, 1) can be writtenas a continued fraction of the form

x =1

a1 +1

a2 +1

a3 +1

. . .

.

Such an expansion is generated by the map T defined on [0, 1) by

Tx =1

x−⌊

1

x

⌋;

see Figure 1.1. The digits an are defined by an = an(x) = a1(Tn−1x) (here

0 1

1

•••

12

13

14

15

16

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 1.1: The continued fraction map T .

T n = T ◦ T ◦ · · · ◦ T ), where

a1 = a1(x) =

{1 if x ∈ (1

2, 1)

n if x ∈ ( 1n+1

, 1n], n ≥ 2.

3

After n iteration of the map T one gets

x =1

a1 + Tx= · · · = 1

a1 +1

a2 +.. . +

1

an + T nx

.

Letpnqn

=1

a1 +1

a2 +.. . +

1

an

,

we will now give some basic properties of this sequence and show thatx = limn→∞

pnqn

. This done by studying elementary properties of matricesassociated with the continued fraction expansion.

Let A ∈ SL2(Z), that is

A =

[r ps q

],

where r, s, p, q ∈ Z and det A = rq− ps ∈ {±1}. Now define the Mobius (or:fractional linear) transformation A : C∗ → C∗ by

A(z) =

[r ps q

](z) =

rz + p

sz + q.

Let a1, a2, . . . be the sequence of partial quotients of x. Put

An :=

[0 11 an

], n ≥ 1 (1.1)

andMn := A1A2 · · ·An , n ≥ 1.

Writing

Mn :=

[rn pnsn qn

], n ≥ 1,

it follows from Mn = Mn−1An, n ≥ 2, that[rn pnsn qn

]=

[rn−1 pn−1sn−1 qn−1

] [0 11 an

],

4

yielding the recurrence relations

p−1 := 1; p0 := a0 = 0; pn = anpn−1 + pn−2 , n ≥ 1,

q−1 := 0; q0 := 1; qn = anqn−1 + qn−2 , n ≥ 1.(1.2)

Furthermore, pn(x) = qn−1(Tx) for all n ≥ 0, and x ∈ (0, 1).Now

ωn = Mn(0) =pnqn

and from det Mn = (−1)n it follows, that

pn−1qn − pnqn−1 = (−1)n , n ≥ 1, (1.3)

hencegcd(pn, qn) = 1 , n ≥ 1,

Setting

A∗n :=

[0 11 an + T nx

],

it follows from

x = Mn−1A∗n(0) = [ 0; a1, . . . , an−1, an + T nx ],

that

x =pn + pn−1T

nx

qn + qn−1T nx, (1.4)

i.e., x = Mn(T nx). From this and (1.3) one at once has that

x− pnqn

=(−1)n T nx

qn(qn + qn−1T nx). (1.5)

In fact, (1.5) yields information about the quality of approximation of therational number ωn = pn/qn to the irrational number x. Since T nx ∈ [0, 1),it at once follows that ∣∣∣∣x− pn

qn

∣∣∣∣ < 1

q2n, n ≥ 0. (1.6)

From 1/T nx = an+1 + T n+1x one even has

1

2qnqn+1

<

∣∣∣∣x− pnqn

∣∣∣∣ < 1

qnqn+1

, n ≥ 1.

5

Notice that the recurrence relations (1.3) yield that

ωn − ωn−1 =(−1)n+1

qn−1qn, n ≥ 1. (1.7)

From this and (1.5) one sees, that

0 = ω0 < ω2 < ω4 < · · ·x · · · < ω3 < ω1 < 1 . (1.8)

In view of proposition 1 the following questions arise naturally: “Given asequence of positive integers (an)n≥1, does limn→∞ ωn exist? Moreover, incase the limit exists and equals x, do we have that x = [ 0; a1, . . . , an, . . . ]?”We have the following proposition.

Proposition 1.1.1 Let (an)n≥1 be a sequence of positive integers, and letthe sequence of rational numbers (ωn)n≥1 be given by

ωn := [ 0; a1, . . . , an ] , n ≥ 1.

Then there exists an irrational number x for which

limn→∞

ωn = x

and we moreover have that x = [ 0; a1, a2, . . . , an, . . . ].

Proof. Writing ωn = pn/qn, n ≥ 1, ω0 := 0, where[pn−1 pnqn−1 qn

]= A1 · · ·An

and where Ai is defined as in (1.1), one has from (1.7), (1.8) and ω0 := 0that

ωn =n∑

k=1

(−1)k+1

qk−1qk,

hence Leibniz’ theorem yields that limn→∞ ωn exists and equals, say, x. Inorder to show that the sequence of positive integers (an)n≥1 determines aunique x ∈ R we have to show that an = an(x) for n ≥ 1, i.e. that (an)n≥1 isthe sequence of partial quotients of x. Since

ωn = [ 0; a1, a2, . . . , an ] =1

a1 + [ 0; a2, a3, . . . , an ]

6

it is sufficient to show that ⌊1

x

⌋= a1 .

However,

ωn =1

a1 + ω∗n,

where ω∗n = [ 0; a2, a3, . . . , an ]. Hence taking limits n→∞ yields

x =1

a1 + x∗,

here x∗ = limn→∞ ω∗n. From 0 < ω∗2 < x∗ < ω∗3 < 1, see also (1.8), and from

1/x = a1 + x∗ it now follows that b1/xc = a1. �

Exercise 1.1.1 Let

∆(a1, . . . , an) := {x ∈ [0, 1) : a1(x) = a1, . . . , an(x) = an} ,

where aj ∈ N for each 1 ≤ j ≤ n. Show that ∆(a1, a2, . . . , ak) is an intervalin [0, 1) with endpoints

pkqk

andpk + pk−1qk + qk−1

,

wherepnqn

=1

a1 +1

a2 +.. . +

1

an

.

1.2 Ergodic Theory setup

The fact that the Continued fraction expansion is generated by the iterationof the map T defined at the beginning of section (1.1) allows us to view thingsin the setup of Ergodic theory. For this we need few ingredients.

We start with a probability space (X,B, µ), and a measurable map T :X → X (so for all B ∈ B we have T−1(B) ∈ B). The map T can be thoughtof as describing the evolution of the system (X,B) after one unit of time,and T n = T ◦ T ◦ · · ·T describes the evolution after n units of time. Wewould like the evolution to be stationary. In the language of ergodic theory,we want T to be measure preserving.

7

Definition 1.2.1 Let (X,B, µ) be a probability space, and T : X → X mea-surable. The map T is said to be measure preserving with respect to µ (orthat µ is T -invariant) if µ(T−1A) = µ(A) for all A ∈ B.

In addition to T -invariance, we need an extra property which is a weakform of independence.

Definition 1.2.2 Let T be a measure preserving transformation on a proba-bility space (X,F , µ). The map T is said to be ergodic if for every measurableset A satisfying T−1A = A, we have µ(A) = 0 or 1.

Theorem 1.2.1 Let (X,F , µ) be a probability space and T : X → X mea-sure preserving. The following are equivalent:

(i) T is ergodic.

(ii) If B ∈ F with µ(T−1B∆B) = 0, then µ(B) = 0 or 1.

(iii) If A ∈ F with µ(A) > 0, then µ (∪∞n=1T−nA) = 1.

(iv) If A,B ∈ F with µ(A) > 0 and µ(B) > 0, then there exists n > 0 suchthat µ(T−nA ∩B) > 0.

Remark 1.2.11. In case T is invertible, then in the above characterization one can replaceT−n by T n.

2. Note that if µ(B4T−1B) = 0, then µ(B \ T−1B) = µ(T−1B \ B) = 0.Since

B =(B \ T−1B

)∪(B ∩ T−1B

),

and

T−1B =(T−1B \B

)∪(B ∩ T−1B

),

we see that after removing a set of measure 0 from B and a set of measure 0from T−1B, the remaining parts are equal. In this case we say that B equalsT−1B modulo sets of measure 0.

3. In words, (iii) says that if A is a set of positive measure, almost everyx ∈ X eventually (in fact infinitely often) will visit A.

4. (iv) says that elements of B will eventually enter A.

8

Proof of Theorem 1.2.1.

(i)⇒ (ii) Let B ∈ F be such that µ(B∆T−1B) = 0. We shall define ameasurable set C with C = T−1C and µ(C∆B) = 0. Let

C = {x ∈ X : T nx ∈ B i.o. } =∞⋂n=1

∞⋃k=n

T−kB.

Then, T−1C = C, hence by (i) µ(C) = 0 or 1. Furthermore,

µ(C∆B) = µ

(∞⋂n=1

∞⋃k=n

T−kB ∩Bc

)+ µ

(∞⋃n=1

∞⋂k=n

T−kBc ∩B

)

≤ µ

(∞⋃k=1

T−kB ∩Bc

)+ µ

(∞⋃k=1

T−kBc ∩B

)

≤∞∑k=1

µ(T−kB∆B

).

Using induction (and the fact that µ(E∆F ) ≤ µ(E∆G)+µ(G∆F )), one canshow that for each k ≥ 1 one has µ

(T−kB∆B

)= 0. Hence, µ(C∆B) = 0

which implies that µ(C) = µ(B). Therefore, µ(B) = 0 or 1.

(ii)⇒ (iii) Let µ(A) > 0 and let B =⋃∞

n=1 T−nA. Then T−1B ⊂ B. Since T

is measure preserving, then µ(B) > 0 and

µ(T−1B∆B) = µ(B \ T−1B) = µ(B)− µ(T−1B) = 0.

Thus, by (ii) µ(B) = 1.

(iii)⇒ (iv) Suppose µ(A)µ(B) > 0. By (iii)

µ(B) = µ

(B ∩

∞⋃n=1

T−nA

)= µ

(∞⋃n=1

(B ∩ T−nA)

)> 0.

Hence, there exists k ≥ 1 such that µ(B ∩ T−kA) > 0.

(iv)⇒ (i) Suppose T−1A = A with µ(A) > 0. If µ(Ac) > 0, then by (iv)there exists k ≥ 1 such that µ(Ac ∩ T−kA) > 0. Since T−kA = A, it followsthat µ(Ac ∩ A) > 0, a contradiction. Hence, µ(A) = 1 and T is ergodic. �

The following lemma provides, in some cases, a useful tool to verify that ameasure preserving transformation defined on ([0, 1),B, µ) is ergodic, whereB is the Lebesgue σ-algebra, and µ is a probability measure equivalent toLebesgue measure λ (i.e., µ(A) = 0 if and only if λ(A) = 0).

9

Lemma 1.2.1 (Knopp’s Lemma) If B is a Borel set and C is a class ofsubintervals of [0, 1), satisfying

(a) every open subinterval of [0, 1) is at most a countable union of disjointelements from C,

(b) ∀A ∈ C , λ(A ∩B) ≥ γλ(A), where γ > 0 is independent of A,

then λ(B) = 1.

Proof. The proof is done by contradiction. Suppose λ(Bc) > 0. Given ε > 0,since the Borel σ-algebra is generated by open intervals there exists a setEε that is a finite disjoint union of open intervals such that λ(Bc4Eε) < ε.Now by conditions (a) and (b) (that is, writing Eε as a countable union ofdisjoint elements of C) one gets that λ(B ∩ Eε) ≥ γλ(Eε).

Also from our choice of Eε and the fact that

λ(Bc4Eε) ≥ λ(B ∩ Eε) ≥ γλ(Eε) ≥ γλ(Bc ∩ Eε) > γ(λ(Bc)− ε),

we have thatγ(λ(Bc)− ε) < λ(Bc4Eε) < ε,

implying that γλ(Bc) < ε+ γε. Since ε > 0 is arbitrary, we get a contradic-tion. �

A central result in Ergodic theory is the following result which can beseen as a generalization of the Strong law of numbers in Probability theory.

Theorem 1.2.2 (The Ergodic Theorem) Let (X,F , µ) be a probability spaceand T : X → X a measure preserving and ergodic transformation. Then, forany f in L1(µ) and for µ a.e. x, one has

limn→∞

1

n

n−1∑i=0

f(T i(x)) =

∫X

f dµ.

As an application of the Ergodic theorem, suppose one is interested ina particular property of the ergodic system (X,B, µ) described by some setA ∈ B. Starting with a point x ∈ X, we look at the orbit or trajectory ofx, namely the set {x, Tx, T 2x, · · · }. We would like to know the frequencyof times that the orbit of x possesses property A. We can formalize this

10

question as follows. Let 1A denotes the indictor function of the set A, namely1A(x) = 1 if x ∈ A and 0 otherwise. For each n,

1

n#{0 ≤ i ≤ n− 1 : T ix ∈ A} =

1

n

n−1∑i=0

1A(T ix),

which is an ergodic average. Applying the Ergodic theorem, we see that

limn→∞

1

n#{0 ≤ i ≤ n−1 : T ix ∈ A} = lim

n→∞

1

n

n−1∑i=0

1A(T ix) =

∫X

1A dµ = µ(A).

Using the Ergodic theorem, one can give a new characterization of ergod-icity as asymptotic average independence.

Corollary 1.2.1 Let (X,F , µ) be a probability space, and T : X → X ameasure preserving transformation. Then, T is ergodic if and only if for allA,B ∈ F , one has

limn→∞

1

n

n−1∑i=0

µ(T−iA ∩B) = µ(A)µ(B). (1.9)

1.3 Continued fractions viewed ergodically

Now consider the continued fraction map Tx =1

x−⌊

1

x

⌋generating the

continued fraction expansion. To view it in the setup of ergodic theory, weneed to find a T -invariant ergodic measure. This invariant measure was foundby Gauss in 1800, and is known nowadays as the Gauss measure which isgiven by

µ(A) =1

log 2

∫A

1

1 + xdx

for all Borel sets A ⊂ [0, 1), where log refers to the natural logarithm. Toprove measurablity of T and the T -invariance of µ, it is enough to check onintervals of the form [a, b) since these generate the Borel σ-algebra on [0, 1).Easy calculation shows that

T−1[a, b) =∞⋃n=1

(1

n+ b,

1

n+ a

],

11

and µ([a, b)) = µ(T−1[a, b)).

Ergodicity is proved by Knopp’s Lemma. We first define the notion of fun-damental intervals. A fundamental interval of order n is a set of the form

∆(a1, . . . , an) := {x ∈ [0, 1) : a1(x) = a1, . . . , an(x) = an} ,

where aj ∈ N for each 1 ≤ j ≤ n. When a1, . . . , an are fixed, we sometimeswrite ∆n instead of ∆(a1, a2, . . . , an). We list few properties mentioned insection 1.1.

(i) ∆(a1, a2, . . . , ak) is an interval in [0, 1) with endpoints

pkqk

andpk + pk−1qk + qk−1

,

wherepnqn

=1

a1 +1

a2 +.. . +

1

an

.

(ii) The sequences (pn)n≥−1 and (qn)n≥−1 satisfy the following recurrence.

p−1 := 1; p0 := a0; pn = anpn−1 + pn−2 , n ≥ 1,

q−1 := 0; q0 := 1; qn = anqn−1 + qn−2 , n ≥ 1.

Furthermore, pn(x) = qn−1(Tx) for all n ≥ 0, and x ∈ (0, 1).

If ai = 1 for all i, one gets the continued fraction expansion of the

small golden mean g =

√5− 1

2. The recursion relations imply that the

denominators qn associated with the golden mean satisfy

q0 = 1, q1 = 1 and qn = qn−1 + qn−1, n ≥ 1.

Thus, the denominators of g are given by the Fibonacci sequence (Fn).This is the slowest sequence of denominators, so for any irrational x,we have qn(x) ≤ Fn.

(iii)

λ (∆(a1, a2, . . . , ak)) =1

qk(qk + qk−1).

12

(iv) If 0 ≤ a < b ≤ 1, that {x : a ≤ T nx < b} ∩∆n equals[pn−1a+ pnqn−1a+ qn

,pn−1b+ pnqn−1b+ qn

)when n is even, and equals(pn−1b+ pn

qn−1b+ qn,pn−1a+ pnqn−1a+ qn

]for n odd. Here ∆n = ∆n(a1, . . . , an) is a fundamental interval of rankn. This leads to

λ(T−n[a, b) ∩∆n) = λ([a, b))λ(∆n)qn(qn−1 + qn)

(qn−1b+ qn)(qn−1a+ qn).

Since

1

2<

qnqn−1 + qn

<qn(qn−1 + qn)

(qn−1b+ qn)(qn−1a+ qn)<

qn(qn−1 + qn)

q2n< 2 .

Therefore we find for every interval I, that

1

2λ(I)λ(∆n) < λ(T−nI ∩∆n) < 2λ(I)λ(∆n) .

Let A be a finite disjoint union of such intervals I. Since Lebesguemeasure is additive one has

1

2λ(A)λ(∆n) ≤ λ(T−nA ∩∆n) ≤ 2λ(A)λ(∆n) . (1.10)

The collection of finite disjoint unions of such intervals generates theBorel σ-algebra. It follows that (1.10) holds for any Borel set A.

(v) For any Borel set A one has

1

2 log 2λ(A) ≤ µ(A) ≤ 1

log 2λ(A) , (1.11)

hence by (1.10) and (1.11) one has

µ(T−nA ∩∆n) ≥ log 2

4µ(A)µ(∆n) . (1.12)

13

Now let C be the collection of all fundamental intervals ∆n. Since theset of all endpoints of these fundamental intervals is the set of all rationalsin [0, 1), it follows that condition (a) of Knopp’s Lemma is satisfied. Nowsuppose that B is invariant with respect to T and µ(B) > 0. Then it followsfrom (1.12) that for every fundamental interval ∆n

µ(B ∩∆n) ≥ log 2

4µ(B)µ(∆n) .

So condition (b) from Knopp’s Lemma is satisfied with γ = log 24µ(B); thus

µ(B) = 1; i.e. T is ergodic. �

Let us exploit the ergodicity of the continued fraction map to reprovesome old results of Paul Levy.

Proposition 1.3.1 For each x ∈ [0, 1) consider the sequence of digits of x

defined by an(x) = an = b 1

T n−1xc, and let λ denote the normalized Lebesgue

measure on [0, 1). Then for λ a.e. x, one has

limn→∞

(a1a2 . . . an)1/n =∞∏k=1

(1 +

1

k(k + 2)

) log klog 2

,

and

limn→∞

a1 + a2 + · · ·+ ann

=∞.

Proof. Let f(x) = log a1(x), then f(T n−1x) = an(x) for any x ∈ (0, 1). Wefirst show that f ∈ L1(µ).∫

f dµ =1

log 2

∫ 1

0

log a1(x)

1 + xdx

=1

log 2

∞∑n=1

∫ 1n

1n+1

log n

1 + xdx

=1

log 2

∞∑n=1

log n · log[1 +1

n(n+ 1)]

≤ 1

log 2

∞∑n=1

log n · 1

n(n+ 1).

14

The last inequality follows from the fact that log(1 +x) < x for x > 0. Now,notice that log x ≤

√x for x ≥ 5, so that log n ≤

√n for all n ≥ 5. Since

∞∑n=5

√n

n(n+ 2)<∞,

it follows that1

log 2

∞∑n=1

log n · 1

n(n+ 1)<∞.

Hence f ∈ L1(µ). By the ergodic theorem

limn→∞

1

n

n∑i=1

log ai(x) = limn→∞

n−1∑i=0

f(T ix) =

∫f dµ =

∞∑n=1

log[1 +1

n(n+ 1)]lognlog 2 ,

µ and hence λ a.e. Thus,

limn→∞

(a1a2 . . . an)1/n =∞∏k=1

(1 +

1

k(k + 2)

) log klog 2

λ a.e.

For the second limit, we consider the function g(x) = a1(x), and notice theg(x) ≥ 1

x− 1 = 1−x

xfor all x ∈ (0, 1). We first show that g /∈ L1(µ). Now∫g dµ ≥ 1

log 2

∫ 1

0

1− xx(x+ 1)

dx

=1

log 2

∫ 1

0

1

x− 2

(x+ 1)dx

= − lim↓0

log

[x

(1 + x)2

]=∞.

Thus, g /∈ L1(µ) so that the ergodic theorem cannot be directly applied tothis function. To surpass this problem, we will truncate g as follows. LetgN(x) = g(x)1[ 1

N+1,1). Then, gN(x) ≤ N , and gN ∈ L1(µ). Furthermore,

{gN} is an increasing sequence converging to g. By Beppo-Levi,

limN→∞

∫gN dµ =

∫g dµ =∞.

15

Now, for each N , we have

lim infn→∞

1

n

n∑i=1

ai(x) = lim infn→∞

1

n

n∑i=0

g(T ix)

≥ lim infn→∞

1

n

n∑i=0

gN(T ix)

= limn→∞

1

n

n∑i=0

gN(T ix)

=

∫gN dµ,

µ and hence λ a.e. Thus,

lim infn→∞

1

n

n∑i=1

ai(x) ≥ limN→∞

∫gN dµ =∞

λ a.e. This implies the result.

Proposition 1.3.2 For almost all x ∈ [0, 1) one has

limn→∞

1

nlog qn =

π2

12 log 2, (1.13)

limn→∞

1

nlog(λ(∆n)) =

−π2

6 log 2, and (1.14)

limn→∞

1

nlog |x− pn

qn| =

−π2

6 log 2. (1.15)

Proof. By the recurrence relations (1.3), for any irrational x ∈ [0, 1) one has

1

qn(x)=

1

qn(x)

pn(x)

qn−1(Tx)

pn−1(Tx)

qn−2(T 2x)· · · p2(T

n−2x)

q1(T n−1x)

=pn(x)

qn(x)

pn−1(Tx)

qn−1(Tx)· · · p1(T

n−1x)

q1(T n−1x).

Taking logarithms yields

− log qn(x) = logpn(x)

qn(x)+ log

pn−1(Tx)

qn−1(Tx)+ · · ·+ log

p1(Tn−1x)

q1(T n−1x). (1.16)

16

For any k ∈ N, and any irrational x ∈ [0, 1), pk(x)qk(x)

is a rational number

close to x. Therefore we compare the right-hand side of (1.16) with

log x+ log Tx+ log T 2x+ · · ·+ log(T n−1x) .

We have

− log qn(x) = log x+ log Tx+ log T 2x+ · · ·+ log(T n−1x) +R(n, x) .

In order to estimate the error term R(n, x), we recall from Exercise 1.1.1that x lies in the interval ∆n, which has endpoints pn/qn and (pn+pn−1)/(qn+qn−1). Therefore, in case n is even, one has

| log x− logpnqn| = log x− log

pnqn

= (x− pnqn

)1

ξ≤ 1

qn(qn−1 + qn)

1

pn/qn<

1

qn,

where ξ ∈ (pn/qn, x) is given by the mean value theorem. Let F1, F2, · · ·be the sequence of Fibonacci 1, 1, 2, 3, 5, . . . (these are the qi’s of the small

golden ratio g√5−12

). It follows from the recurrence relation for the qi’s thatqn(x) ≥ Fn. A similar argument shows that

| log x− logpnqn| < 1

qn

in case n is odd. Thus

|R(n, x)| ≤ 1

Fn

+1

Fn−1+ · · ·+ 1

F1

,

and since we have

Fn =Gn + (−1)n+1gn√

5

it follows that Fn ∼ 1√5Gn, n → ∞. Thus 1

Fn+ 1

Fn−1+ · · · + 1

F1is the nth

partial sum of a convergent series, and therefore

|R(n, x)| ≤ 1

Fn

+ · · ·+ 1

F1

≤∞∑n=1

1

Fn

:= C.

Hence for each x for which

limn→∞

1

n(log x+ log Tx+ log T 2x+ · · ·+ log(T n−1x))

17

exists,

− limn→∞

1

nlog qn(x)

exists too, and these limits are equal.

Now limn→∞

1

n(log x+log Tx+log T 2x+· · ·+log(T n−1x)) is ideally suited for

the Pointwise Ergodic Theorem; we only need to check that the conditionsof the Ergodic Theorem are satisfied and to calculate the integral. This isleft as an exercise for the reader. This proves (1.13).

We have seen that

λ(∆n(a1, . . . , an)) =1

qn(qn + qn−1);

thus

− log 2− 2 log qn < log λ(∆n) < −2 log qn .

Now apply (1.13) to obtain (1.14). Finally (1.15) follows from (1.13) and

1

2qnqn+1

<

∣∣∣∣x− pnqn

∣∣∣∣ < 1

qnqn+1

, n ≥ 1.

�

1.4 Approximation coefficients

We define for every real number x ∈ [0, 1) and every n ≥ 0 the so-calledapproximation coefficients Θn(x) by

Θn(x) = q2n

∣∣∣∣x− pnqn

∣∣∣∣ . (1.17)

From equation 1.5 in section 1.1 we have

Θn = q2n

∣∣∣∣x− pnqn

∣∣∣∣ =T n(x)

1 + T n(x) qn−1

qn

.

18

By the recurrence-relations (1.3) we have

qn−1qn

=qn−1

anqn−1 + qn−2=

1

an + qn−2

qn−1

=1

an +1

an−1 +.. . +

1

a1

.

Thus, Θn depends on the future T nx = 1

an+1 +1

an+2 +1

an+3 +1

. . .

, and the

pastqn−1qn

=1

an +1

an−1 +.. . +

1

a1

. Define an invertible map T : [0, 1) ×

[0, 1]→ [0, 1)× [0, 1] by

T (x, y) =

(T (x),

1⌊1x

⌋+ y

), x 6= 0, T (0, y) = (0, y).

So if x = 1

a1 +1

a2 +1

a3 +1

. . .

, then

T (x, 0) = (1

a2 +1

a3 +1

a4 +1

. . .

,1

a1),

T 2(x, 0) = (1

a3 +1

a4 +1

a5 +1

. . .

,1

a2 +1

a1

),

19

and in general

T n(x, 0) = (1

an+1 +1

an+2 +1

an+3 +1

. . .

,1

an +.. .

1

a1

),

or

T n(x, 0) = (1

an+1 +1

an+2 +1

an+3 +1

. . .

,qn−1qn

).

If we define f : [0, 1) × [0, 1) → [0, 1) by f(x, y) =x

1 + xy, then Θn(x) =

f(T n(x, 0)). The map T has a T -invariant ergodic measure µ defined by

µ(E) =1

log 2

∫∫E

1

(1 + xy)2dλ(x) dλ(y).

So by the Ergodic theorem we can say something over the asymptoticaverages of the approximation coefficients, namely

limn→∞

1

n

n∑i=1

Θi(x) = limn→∞

1

n

n∑i=1

Θi(x)

=

∫ 1

0

∫ 1

0

f(x, y) dµ

=1

log 2

∫ 1

0

∫ 1

0

x

(1 + xy)3dy dx =

1

4 log 2.

Another application is the following result known as the Lenstra-DoeblinConjecture. Let z ∈ (0, 1), then for λ a.e. x,

limn→∞

1

n#{j ; 1 ≤ j ≤ n, Θj(x) ≤ z} = F (z) =

z

log 20 ≤ z ≤ 1

2

1

log 2(1− z + log 2z) 1

2≤ z ≤ 1.

20

The proof is a simple application of the Ergodic theorem. To see this, let

Az = {(x, y) ∈ [0, 1)×[0, 1) :x

1 + xy≤ z} = {(x, y) ∈ [0, 1)×[0, 1) : f(x, y) ≤ z}.

Then, Θi(x) = f(T i(x, 0) ≤ z ⇔ T i(x, 0) ∈ Az ⇔ 1Az(T i(x, 0) = 1, so

limn→∞

1

n#{j ; 1 ≤ j ≤ n, Θj(x) ≤ z} = lim

n→∞

1

n

n∑i=1

1Az(T i(x, 0) = µ(Az) = F (z).

Bibliography

[Bil] Billingsley, P. – Ergodic Theory and Information, John Wiley andSons, 1965.

[Bor] Borel, E. – Contribution a l’analyse arithmetiques du continu, J.Math. Pures et Appl. (5) 9 (1903), 329-375.

[BJW] Bosma, W., H. Jager and F. Wiedijk – Some metrical observations onthe approximation by continued fractions, ibid. 45 (1983), 281-270.

[BK] Bosma, W. and C. Kraaikamp – Metrical theory for optimal continuedfractions, J. Number Theory 34 (1990), 251-270.

[BS] Brin, M., and Stuck, G., Introduction to Dynamical Systems, Cam-bridge University Press, 2002.

[CFS] Cornfeld, I. P., S. V. Fomin and Ya. G. Sinai – Ergodic Theory,Springer-Verlag, 1981.

[DK] Dajani, K., and Kraaikamp, C., Ergodic theory of numbers, CarusMathematical Monographs, 29. Mathematical Association of Amer-ica, Washington, DC 2002.

[De] Devaney, R.L., An Introduction to Chaotic Dynamical Systems, Sec-ond Edition, Addison-Wesley Publishing Company, Inc., 1989.

[Doe] Doeblin, W. – Remarques sur la theorie metrique des fractions con-tinues, Comp. Math. 7, (1940), 353-371.

[Ge] Gel’fond, A.O. – A common property of number systems, Izv. Akad.Nauk SSSR. Ser. Mat. 23 (1959) 809–814.

21

22

[Jag1] Jager, H. – On the speed of convergence of the nearest integer contin-ued fraction, Math. Comp. 39, (1982), 555-558.

[IK] Iosifescu, I., and Kraaikamp, C., Metrical theory of continued frac-tions. Mathematics and its Applications, 547. Kluwer Academic Pub-lishers, Dordrecht, 2002.

[Jag2] Jager, H. – The distribution of certain sequences connected with thecontinued fraction, Indag. Math. 48, (1986), 61-69.

[JK] Jager, H. and C. Kraaikamp – On the approximation by continuedfractions, Indag. Math., 51 (1989), 289-307.

[Knu2] Knuth, D.E. – The distribution of continued fraction approximations,J. Number Theory 19, (1984), 443-448.

[KN] Kuipers, L. and H. Niederreiter – Uniform Distribution of Sequences,John Wiley and Sons, 1974.

[KK] Kamae, T., and Keane, M.S., A simple proof of the ratio ergodictheorem, Osaka J. Math. 34 (1997), no. 3, 653–657.

[Knu] Knuth, D.E. – The distribution of continued fraction approximations,J. Number Theory 19, (1984), 443-448.

[KT] Kingman and Taylor, Introduction to measure and probability, Cam-bridge Press, 1966.

[L] Luroth, J. – Ueber eine eindeutige Entwickelung von Zahlen in eineunendliche Reihe, Math. Annalen 21 (1883), 411-423.

[Le] Levy, P. – Sur le loi de probabilite dont dependent les quotients com-plets wet incompletes d’une fraction continue, Bull. Soc. Math. deFrance 57, (1929), 178-194.

[Min] Minkowski, H. – Uber die Annaherung an eine reelle Grosse durchrationale Zahlen, Math. Ann. 54, (1901), 91-124.

[Nak] Nakada, H. – Metrical theory for a class of continued fraction trans-formations and their natural extensions, Tokyo J. Math. 4, (1981),399-426.

23

[Obr] Obrechkoff, N. – Sur l’approximation des nombres irrationels par desnombres rationels, C. R. Acad. Bulgare Sci. 3 (1), (1951), 1-4.

[Pa] Parry, W., Topics in Ergodic Theory, Reprint of the 1981 original.Cambridge Tracts in Mathematics, 75. Cambridge University Press,Cambridge, 2004.

[Per] Perron, O. – Die Lehre von den Kettenbruchen, Band I, 3. verb. u.etw. Aufl., B. G. Teubner, Stuttgart.

[P] Petersen, K., Ergodic Theory, Cambridge Studies in Advanced Math-ematics, 2. Cambridge University Press, Cambridge, 1989.

[Rie1] Rieger, G. J. – Ein Gauss-Kusmin-Levy Satz fur Kettenbruche nachnachsten Ganzen, Manuscripta Math. 24, (1978), 437-448.

[Rie2] Rieger, G. J. – Uber die Lange von Kettenbruchen mit ungeradenTeilnennern, Abh. Braunschweig Wiss. Ges. 32, (1981), 61-69.– On the metrical theory of the continued fractions with odd partialquotients, in: Topics in Classical Number Theory, vol. II, Budapest1981, 1371-1481 (Colloq. Math. Soc. Janos Bolyai, 34, North-Holland1984).

[RN] Ryll-Nardzewski, C. – On the ergodic theorems (II) (Ergodic theoryof continued fractions), Studia Math. 12, (1951), 74-79.

[S1] Schweiger, F. – Continued fractions with odd and even partial quo-tients, Arbeitsbericht Math. Inst. Univ. Salzburg 4, (1982), 59-70.– On the approximation by continued fractions with odd and even par-tial quotients, Arbeitsbericht Math. Inst. Univ. Salzburg 1-2, (1984),105-114.

[S2] Schweiger, F. – Ergodic Theory of Fibered Systems and Metric Num-ber Theory, Clarendon Press, Oxford 1995.

[Sen] Sendov, B. – Der Vahlensche Satz uber die singularen Kettenbrucheund die Kettenbruche nach nachsten Ganzen, Ann. Univ. Sofia, Fac.Sci. Math. Livre 1 Math. 54, (1959/60), 251-258.

24

[St] Steuding, J.– Diophantine analysis. Discrete Mathematics and its Ap-plications. Chapman & Hall/CRC, Boca Raton, FL, 2005.

[Tie] Tietze, H. – Uber die raschesten Kettenbruchentwicklung reellerZahlen, Monatsh. Math. Phys. 24, (1913), 209-241.

[W] Walters, P., An Introduction to Ergodic Theory, Graduate Texts inMathematics, 79. Springer-Verlag, New York-Berlin, 1982.