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### Master’s Thesis

## Continued Fraction Expansions of Two Close Irrational Numbers

### Yunjeong Oh

### Department of Mathematical Sciences

### Ulsan National Institute of Science and Technology

### 2023

## Continued Fraction Expansions of Two Close Irrational Numbers

### Yunjeong Oh

### Department of Mathematical Sciences

### Ulsan National Institute of Science and Technology

### Abstract

For an irrational number*x*in(0*,*1), we consider its approximation by continued fraction expansion and
by decimal expansion. Comparing the two approximations,*k**n*(*x*)is deﬁned by the number of digits of
continued fraction expansion of*x*that are determined by those of the*n*-th decimal approximation of*x*,
for any*n≥*1. G. Lochs proved in 1964 that the ratio ^{k}^{n}_{n}^{(}^{x}^{)} converges to a constant 6 log 2 log 10

*π*^{2} as*n→*∞
(see [5]). The observable point is that we can prove the theorem without using the decimal expansion.

Focusing on this fact, we drop the base notation and take ^{}^{Nx}_{Nx}^{} as an approximation of*x*for any integer
*N*. This approximation converges to*x* as*N* increases like the*n*-th decimal expansion approximation.

Therefore we deﬁne a quantity*K**N*(*x*)in a similar sense, comparing the continued fraction approximation
with ^{}^{Nx}_{Nx}^{}. In this thesis, we restate the theorem of Lochs in terms of*K**N*(*x*)and log*N* and prove the
renewed statement. Furthermore we restate and reprove several results about the distribution of ^{K}_{log}^{N}^{(}_{N}^{x}^{)}
given by C. Faivre when*N*=10^{n}.

### Contents

I Introduction . . . 1

II Preliminaries . . . 3

2.1 Basic Properties of Continued Fractions . . . 3

2.2 Ergodic Theory and Lévy Constant . . . 5

2.3 A Central Limit Theorem for the sequence*k**n*(*x*) . . . 11

2.4 Transfer Operators and Measure of Exceptional Set . . . 12

III *N*-approximation version of Theorem of Lochs . . . 16

IV *N*-approximation version of Central Limit Theorem . . . 21

V *N*-approximation version of Conditional Probability of Exceptional Set . . . 24

References . . . 29

Acknowledgements . . . 30

### I Introduction

For a real number*x*, consider an expression

*x*=*a*0+ 1
*a*1+_{a} ^{1}

2+_{a}_{3+···}^{1}

for*a*0*,a*1*,a*2*,··· ∈*N. Such expression is called the continued fraction of*x*, denoted by[*a*0;*a*1*,a*2*,···*].
For*k≥*0, we write ^{p}_{q}^{k}

*k* = [0;*a*1*,a*2*,···,a**k*], which is called the*k*-th convergent of[*a*0;*a*1*,a*2*,···*]. Each
*a**i*is called the the*i*-th digit of the continued fraction.

If there is a positive integer*n*such that*x*= [*a*0;*a*1*,···,a**n*], we call it a ﬁnite continued fraction. If not,
we call it an inﬁnite continued fraction. It is well known that a rational number has a ﬁnite continued
fraction expansion and an irrational number has an inﬁnite continued fraction expansion that converges
to itself. Hence for an irrational number *x∈*[0*,*1], we have its inﬁnite continued fraction expansion
*x*= [0;*a*1*,a*2*,···*]. Then ^{p}_{q}^{k}

*k* converges to*x*as*k→*∞, so we obtain an approximation of*x*. On the other
hand, we have another approximation of*x*, the decimal expansion of*x*. More precisely, consider an*n*-th
decimal approximation*d**n*(*x*)and*e**n*(*x*)of*x*, which are given as

*d**n*(*x*) =10^{n}*x*

10^{n}*x* and *e**n*(*x*) =10^{n}*x*+1
10^{n}*x* *.*

Since*d**n*(*x*)and*e**n*(*x*)are rational numbers, they have continued fraction expansions. Denoting the*i*-th
digit of*x*by*a**i*(*x*), write the continued fraction expansion of*d**n*(*x*)and*e**n*(*x*)as:

*d**n*(*x*) = [0;*a*1(*d**n*(*x*))*,a*2(*d**n*(*x*))*,a*3(*d**n*(*x*))*,...,a**k*(*d**n*(*x*))*,...*]
*e**n*(*x*) = [0;*a*1(*e**n*(*x*))*,a*2(*e**n*(*x*))*,a*3(*e**n*(*x*))*,...,a**k*(*e**n*(*x*))*,...*]
Now deﬁne*k**n*(*x*)as

*k**n*(*x*) =max*{k≥*0*|a**i*(*d**n*(*x*)) =*a**i*(*e**n*(*x*)) for all 0*≤i≤k}*

and denote*a**i*=*a**i*(*d**n*(*x*))=*a**i*(*e**n*(*x*))for 0*≤*i*≤k**n*(*x*). Then

*d**n*(*x*) = [0;*a*1*,a*2*,a*3*,...,a*_{k}_{n}_{(}_{x}_{)}*,d*_{k}_{n}_{(}_{x}_{)+}_{1}] and

*e**n*(*x*) = [0;*a*1*,a*2*,a*3*,...,a*_{k}_{n}_{(}_{x}_{)}*,e**k*_{n}(*x*)+1]*,* (1)
where1*/d*_{k}_{n}_{(}_{x}_{)+}_{1}* *=1*/e*_{k}_{n}_{(}_{x}_{)+}_{1}by the deﬁnition of*k**n*(*x*). Now, consider a set

*I*(*a*1*,a*2*,·,a**n*) =*{y∈*(0*,*1):*a*1(*y*) =*a*1*,a*2(*y*) =*a*2*,···,a**n*(*y*) =*a**n**}*

for*a*1*,···,a**n**∈*N. It is well-known that*I*(*a*1*,a*2*,·,a**n*)forms an interval in(0*,*1). Then we have

*x∈*[*d**n*(*x*)*,e**n*(*x*)]*⊂I*(*a*1*,a*2*,...,a**k*_{n}(*x*))*,* (2)
which leads us to conclude that

*x*= [0;*a*1*,a*2*,a*3*,...,a**k*_{n}(*x*)*,...*]*.*

Therefore the ﬁrst*k**n*(*x*)digits of continued fraction expansion of*x*are exactly those of*d**n*(*x*)and*e**n*(*x*).
It is natural to ask what kind of relation between*k**n*(*x*)and*n*would have. One easy observation is that as
*n*increases,*k**n*(*x*)must increase without an upper bound so that lim_{n}_{→∞}*k**n*(*x*) =∞. Furthermore, one can
ask how fast*k**n*(*x*)grows as*n*goes to inﬁnity. In other words, we want to see the asymptotic behavior
of ^{k}^{n}_{n}^{(}^{x}^{)} as*n→*∞. Gustav Lochs proved the following results in 1964.

Theorem 1(Theorem of Lochs). *For almost all x,*

*n*lim*→*∞

*k**n*(*x*)

*n* = 6 log 2 log 10

*π*^{2} *≈*0*.*9703
*with respect to the Lebesgue measure.*

One observation is that using decimal notation does not play a crucial role in his proof(see pp105- 114 of [2]). The only effect of decimal notation is the appearance of log 10 in the fraction. Hence it is easy to guess that we can prove the theorem by using other bases and it will cause a mere change.

Furthermore, we may drop the base notation and use even a more elementary way of approximation with
a large number*N*as follows:

For a real number*x*and a positive integer*N*,
*d**N*(*x*) =*Nx*

*Nx* and *e**N*(*x*) =*Nx*+1
*Nx* *.*

We call this the*N*-approximation of*x*. Note that the*N*-approximation of*x*does not mean approximating
*x*with base*N*. It is replacing 10^{n}by an integer*N*. Also, note that the*N*-approximation converges to*x*
as*N*goes to inﬁnity.

Since*d**N*(*x*)and*e**N*(*x*)are rational numbers, each has a ﬁnite continued fraction,
*d**N*(*x*) = [0;*a*1(*d**N*(*x*))*,a*2(*d**N*(*x*))*,a*3(*d**N*(*x*))*,···*]

and

*e**N*(*x*) = [0;*a*1(*e**N*(*x*))*,a*2(*e**N*(*x*))*,a*3(*e**N*(*x*))*,···*]*,*

which are called*N*-expansions of*x*. For*d**N*(*x*)and*e**N*(*x*)we deﬁne*K**N*(*x*)in the same sense. To avoid
confusion, we use*K**N*(*x*)instead of*k**N*(*x*). Note that*k**n*(*x*) =*K*10^{n}(*x*).

This thesis’s main goal is to study the new version of Lochs’ Theorem, written in terms of*K**N*(*x*)and
log*N*and observe the difference between the two approximations of*x*. Furthermore, we introduce more
advanced results about the distribution of ^{k}^{n}^{(}_{n}^{x}^{)}, which are given by C. Faivre in next section. Stating
their*N*-approximation version by writing them in terms of ^{K}_{logN}^{N}^{(}^{x}^{)} and proving them is another main goal
of this thesis.

### II Preliminaries

In this section, we introduce more advanced results about the distribution of ^{k}^{n}_{n}^{(}^{x}^{)}. The theorems
are the results of some dynamical properties of continued fraction. We introduce several well-known
theorems and concepts related to the dynamics of continued fractions, which will be useful in later
sections. Before that, we need to see the basic properties of continued fractions that are used frequently.

2.1 Basic Properties of Continued Fractions

Recall from Section 1 that for a real number*x*with its continued fraction expansion
*x*=*a*0+ 1

*a*1+_{a} ^{1}

2+_{a+}3^{1}+*···*

= [*a*0;*a*1*,a*2*,···*]*.*

For*k≥*0, we have its*k*-th convergent ^{p}_{q}^{k}

*k* = [*a*0;*a*1*,a*2*,···*]. And for any(*a*1*,a*2*,···,a**n*)*∈*N^{n}, we have a
set*I*=*I*(*a*1*,a*2*,···,a**n*), deﬁned by

*I*(*a*1*,···,a**n*) =*{y∈*(0*,*1)*|a*1(*y*) =*a*1*,a*2(*y*) =*a*2*,···,a**n*(*y*) =*a**n**}.*

We are already informed that the set*I*forms an interval in(0*,*1). In this section, we prove some useful
properties of *p**k*,*q**k* ﬁrst, by following the proofs in [1]. Using these, we will prove that the set*I* is an
interval in(0*,*1), ﬁnding its precise form.

Proposition 1. *For k≥*2*,*

*p**k*=*a**k**p**k**−*1+*p**k**−*2 *and* *q**k*=*a**k**q**k**−*1+*q**k**−*2*.* (3)
*Proof.* We prove this by induction on*k*. For*k*=2,

*p*2

*q*2 =*a*0+ 1
*a*1+_{a}^{1}_{2}*,*
hence it is a simple calculation to see that (3) is true for*k*=2.

Now, assume that (3) holds for *k* *<n* and consider the case for *k*= *n*. For a continued fraction
[*a*0;*a*1*,a*2*,...*], write

*p*^{}_{k}

*q*^{}_{k} = [*a*1;*a*2*,...,a**k*]*.*

Then we obtain

*p**n*

*q**n* =*a*0+ 1

*p*^{}_{n−}_{1}
*q*^{}_{n−1}

*,*
which gives

*p**n*=*a*0*p*^{}_{n}_{−}_{1}+*q*^{}_{n}_{−}_{1}
*q**n*=*p*^{}_{n}_{−}_{1}

Then by the inductive hypothesis, we get

*p*^{}_{n}_{−}_{1}=*a**n**p*^{}_{n}_{−}_{2}+*p*^{}_{n}_{−}_{3}
and

*q*^{}_{n}_{−}_{1}=*a**n**q*^{}_{n}_{−}_{2}+*q*^{}_{n}_{−}_{3}*.*
Combined with above equation, we obtain

*p**n*=*a*0(*a**n**p*^{}_{n}_{−}_{2}+*p*^{}_{n}_{−}_{3}) + (*a**n**q*^{}_{n}_{−}_{2}+*q**n**−*3)

=*a**n*(*a*0*p*^{}_{n}_{−}_{2}+*q*^{}_{n}_{−}_{2}) + (*a*0*p*^{}_{n}_{−}_{3}+*q*^{}_{n}_{−}_{3})

=*a**n**p**n**−*1+*p**n**−*2

and

*q**n*=*p*^{}_{n}_{−}_{1}=*a**n**p*^{}_{n}_{−}_{2}+*p*^{}_{n}_{−}_{3}

=*a**n**q**n**−*1+*q**n**−*2*,*
which complete the proof.

Proposition 2. *For any k≥*2*,*

*q**k**p**k**−*1*−p**k**q**k**−*1= (*−*1)^{k}*.* (4)
*Proof.* From Proposition 1, we have

*p**k**q**k**−*1=*a**k**p**k**−*1*q**k**−*1+*p**k**−*2*q**k**−*1

*q**k**p**k**−*1=*a**k**q**k**−*1*p**k**−*1+*q**k**−*2*p**k**−*1*,*

which are obtained by multiplying the ﬁrst and the second formulae in (3) by*q**k**−*1and*p**k**−*1, respectively.

Subtracting the second one by the ﬁrst one, we have

*q**k**p**k**−*1*−p**k**q**k**−*1=*−*(*q**k**−*1*p**k**−*2*−p**k**−*1*q**k**−*2)
for any*k≥*2. Hence

*q**k**p**k**−*1*−p**k**q**k**−*1=*−*(*q**k**−*1*p**k**−*2*−p**k**−*1*q**k**−*2)

= (*−*1)^{2}(*q**k**−*2*p**k**−*3*−p**k**−*2*q**k**−*3)
...

= (*−*1)^{k}*,*
proving the proposition.

Now we are ready to prove that*I*(*a*1*,a*2*,···,a**n*)forms an interval in(0*,*1). The interval is called a
fundamental interval.

Proposition 3. *For any a*1*,···,a**n**∈*N*, the set I*(*a*1*,a*2*,···,a**n*)*forms an interval in*(0*,*1)*. More pre-*
*cisely,*

*I*(*a*1*,···,a**n*) =

⎧⎪

⎨

⎪⎩

*p*_{n}+*p*_{n−}1

*q*_{n}+*q*_{n−}1*,*_{q}^{p}_{n}^{n}

*for odd n*
*p*_{n}

*q*_{n}*,*^{p}_{q}^{n}_{n}^{+}_{+}_{q}^{p}_{n−}^{n−}_{1}^{1}

*for even n*

(5)
*where* ^{p}_{q}^{k}

*k* = [0;*a*1*,a*2*,···,a**n*]*, and its length is* _{q}_{n}_{(}_{q}_{n}_{+}^{1}_{q}_{n−}_{1}_{)}*.*

*Proof.* Deﬁne a function *f*(*z*) = [0;*a*1*,a*2*,···,a**n*+*z*]on *z∈*[0*,*1]. Then observe that*I*(*a*1*,···,a**n*) =
*f*((0*,*1))and *f* is continuous so that*I*(*a*1*,···,a**n*)forms an interval. If*n*is odd, one can easily see that
*f* is decreasing and increasing if*n*is even. Thus we obtain

*I*(*a*1*,a*2*,···,a**n*) =

⎧⎨

⎩

[*f*(1)*,f*(0)] for odd n
[*f*(0)*,f*(1)] for even n*.*
Note that Proposition 1 gives

*f*(1) = [0;*a*1*,···,a**n*+1] = *p**n*+*p**n**−*1

*q**n*+*q**n**−*1

and *f*(0) = [0;*a*1*,···,a**n*] = *p**n*

*q**n**.*
It remains to show that the length of*I* is given by _{q} ^{1}

*n*(*q*_{n}+*q*_{n−1}). By Proposition 2,

*|* *p**n*+*p**n**−*1

*q**n*+*q**n**−*1 *−p**n*

*q**n* *|*=*|q**n**p**n**−*1*−p**n**q**n**−*1*|*
*q**n*(*q**n*+*q**n**−*1)

= 1

*q**n*(*q**n*+*q**n**−*1)*,*
completing the proof.

2.2 Ergodic Theory and Lévy Constant

For an irrational number*x*, we know that its partial quotient ^{p}_{q}^{n}

*n* of continued fraction converges to*x*
as*n→*∞. Measuring how fast the convergence is, one can consider how fast*|x−*^{p}_{q}^{n}_{n} *|*converges to 0.

For an irrational number*x*, we can write*x*= [0;*a*1*,a*2*,...,a**n**−*1*,r**n*]with
*r**n*= [*a**n*;*a**n*+1*,a**n*+2*,...*]*.*

Note that

*r**n*=*a**n*+ 1
*r**n*+1*,*

which implies that*r**n**−a**n**<*1. By Proposition 1 and Proposition 2, one can easily see that

*|x−p**n*

*q**n* *|<* 1

(*q**n**−*1*r**n*+*q**n**−*2)(*q**n**−*1*a**n*+*q**n**−*2) *<* 1
*q**n*2*.*
Thus the growth of *q**n* measures the accuracy of the approximation of*x* by ^{p}_{q}^{n}

*n*. In fact, the following
inequality holds for*n≥*1,

*q**n**≥G*^{n}^{−}^{2}*,*

where*G*is the golden ratio(see [6]). Therefore it is natural to consider the following limit for an irrational
number*x*

*β*(*x*) = lim

*n**→*∞

log*q**n*(*x*)

*n* *,*

when the limit exists and ﬁnite. In that case, the limit is called Lévy constant (p. 513, [6]). P. Lévy
proved that Lévy constant exists and has its value as _{12 log 2}^{π}^{2} for almoast all*x∈*[0*,*1], which is stated as
following (see [4]), and G. Lochs proved his theorem by using the theorem of Lévy.

Theorem 2(Theorem of Lévy). *For almost all x∈*[0*,*1]*,*

*n*lim*→*∞

log*q**n*(*x*)

*n* = *π*^{2}

12 log 2

This theorem is proved by applying a dynamical concept called Ergodic System. In this thesis, we introduce the very basics of Ergodic theory. For more details about Ergodic Theory, see [3].

Deﬁnition 1. *Suppose that*(*X,B,μ*)*is a probability space. The measure space*(*X,B,μ*)*together with*
*a measurable map T* :*X* *→X is called a measure-preserving system if T is measure-preserving with*
*respect to* *μ, i.e.* *μ*(*T*^{−}^{1}(*A*)) =*μ*(*A*) *for any A∈B. The map T is said to be ergodic if T*^{−}^{1}(*A*) =*A*
*for A∈Bimplies that* *μ*(*A*) =0*orμ*(*A*) =1*. A measure-preserving system*(*X,B,μ,T*)*is called an*
*ergodic system if T is ergodic.*

Consider a set*X*= [0*,*1]and the Borel*σ*-algebra*B*on*X*. Let*T* be the Gauss transformation on*X*,
given by*T*(*x*) =^{1}_{x}*− *^{1}_{x}. Note that the map gives a shift map on the continued fraction expansion,

*T*[0;*a*1*,a*2*,···*] = [0;*a*2*,a*3*,···*]*,*
thus

*T*^{k}[0;*a*1*,a*2*,···*] = [0;*a**k*+1*,a**k*+2*,···*]*.*

For*k≥*0, it is well known that*T* preserves that the Gauss measure*μ*, the measure with density_{log 2}^{1} _{1}_{+}^{1}_{x}.
In fact, for an interval[*a,b*]in*X*, we have

*T*^{−}^{1}[*a,b*] = ^{}^{∞}

*k*=1

1
*b*+*k,* 1

*a*+*k*

*.*
Thus

*μ*(*T*^{−}^{1}[*a,b*]) = 1
log 2

### ∑

∞*k*=1

^{1}

*a*+*k*
*b*+*k*1

1
1+*xdx*

= 1

log 2

### ∑

∞*k*=1

log

1+ 1
*a*+*k*

*−*log

1+ 1
*b*+*k*

= 1

log 2 lim

*n**→*∞[log(1+*a*+*n*)*−*log(1+*a*)*−*log(1+*b*+*n*) +log(1+*b*)]

= 1

log 2[log(1+*b*)*−*log(1+*a*)]

= 1

log 2

_{b}

*a*

1

1+*xdx*=*μ*[*a,b*]*.*

Therefore, the Gauss map*T* is measure-preserving with respect to*μ*. Hence(*X,B,μ,T*)is a dynamical
system. In the following theorem, we show that*T* is ergodic so that(*X,B,μ,T*)is an ergodic system.

Theorem 3. *Let X* = (0*,*1)*,Bbe the Borelσ-algebra on X, andμ* *be the Gauss measure onB. Let T*
*be the Gauss map on X. Then*(*X,B,μ,T*)*is an ergodic system.*

*Proof.* It sufﬁces to show that the Gauss map*T* is ergodic, which is equivalent to show that if*A∈B*
with*T*^{−}^{1}(*A*) =*A*, then *μ*(*A*) =0 or *μ*(*A*) =1. For any(*a*1*,a*2*,···,a**n*)*∈*N^{n}, consider a fundamental
interval*I*(*a*1*,a*2*,···,a**n*). Let*A*be an measurable subset in(0*,*1)with*T*^{−}^{1}(*A*) =*A*. Note that it is enough
to show that for any*A∈B*,

*μ*(*T*^{−}^{n}(*A*)*∩I*(*a*1*,a*2*,···,a**n*))*μ*(*A*)*μ*(*I*(*a*1*,a*2*,···,a**n*))*,* (6)
where *fg*means there exists some absolute constants*C*and*D*such that

*C f* *≤g≤D f.*

In fact, recall from Proposition 3 that *m*(*I*(*a*1*,a*2*,···,a**n*)) = _{q}_{n}_{(}_{q}_{n}_{+}^{1}_{q}_{n−}_{1}_{)}, where ^{p}_{q}^{n}

*n* = [0;*a*1*,a*2*,···,a**n*],

*p*_{n−}1

*q*_{n−}1 = [0;*a*1*,a*2*,···,a**n**−*1] and*m* is the Lebesgue measure. Then it is easy to deduce that for any*n*
*m*(*I*(*a*1*,a*2*,···,a**n*)) = _{q}_{n}_{(}_{q}_{n}_{+}^{1}_{q}_{n−}_{1}_{)}*<*_{2}*n−*^{1}2 from Proposition 1. Then we have

*n*lim*→*∞*m*(*I*(*a*1*,a*2*,···,a**n*)) =0*,*

where the convergence is uniform. Hence the collection of subsets *{I*(*a*1*,···,a**n*)*|*(*a*1*,···,a**n*)*∈*N^{n}*}*
generates*B*. Therefore (6) is equivalent to say that*μ*(*A∩B*) =*μ*(*A*)*μ*(*B*)for any*B∈B*. Then, taking
*B*= (0*,*1)*−A*, the claim provides that 0=*μ*(*A*)*μ*((0*,*1)*−A*), which is,*μ*(*A*) =0 or*μ*(*A*) =1.

Now it remains to prove (6). Note that it is enough to show this for an arbitrary interval*A*= [*d,e*]*⊂*
(0*,*1). By Proposition 1,

*u∈I*(*a*1*,···,a**n*) *⇐⇒* *u*= [0;*a*1*,···,a**n*+*T*^{n}(*u*)] = *p**n*+*p**n**−*1*T*^{n}(*u*)
*q**n*+*q**n**−*1*T*^{n}(*u*)
and by proposition 3,

*u∈I*(*a*1*,···,a**n*)*∩T*^{−}^{1}(*A*) *⇐⇒* u is between *p**n*+*p**n**−*1*d*

*q**n*+*q**n**−*1*d* and *p**n*+*p**n**−*1*e*
*q**n*+*q**n**−*1*e.*
Thus we have

*m*(*I*(*a*1*,···,a**n*)*∩T*^{−}^{1}(*A*)) =*|* *p**n*+*p**n**−*1*d*

*q**n*+*q**n**−*1*d* *−p**n*+*p**n**−*1*e*
*q**n*+*q**n**−*1*e* *|*

= (*e−d*) *|p**n**q**n**−*1*−p**n**−*1*q**n**|*
(*q**n*+*q**n**−*1*e*)(*q**n*+*q**n**−*1*d*)

= (*e−d*) 1

(*q**n*+*q**n**−*1*e*)(*q**n*+*q**n**−*1*d*)*.*
Thus

*m*(*I*(*a*1*,···,a**n*)*∩T*^{−}^{n}(*A*)) =*m*(*A*)*m*(*I*(*a*1*,···,a**n*)) *q**n*(*q**n*+*q**n**−*1)
(*q**n*+*q**n**−*1*e*)(*q**n*+*q**n**−*1*d*)
*m*(*A*)*m*(*I*(*a*1*,···,a**n*))*.*

(7)

Noticing that there are following inequalities between the Lebesgue measure*m*and the Gauss measure
*μ*:

*m*(*B*)

2 log 2 *≤μ*(*B*)*≤* *m*(*B*)

log 2 (*B∈B*)*,*
we have proved that*μ*(*I*(*a*1*,···,a**n*)*∩T*^{−}^{n}(*A*))*μ*(*A*)*μ*(*I*(*a*1*,···,a**n*)).

Ergodicity has useful applications on various ﬁelds. In particular, the following well known theorem called Birkhoff’s Ergoodic Theorem plays a crucial part in proving Theorem of Lévy.

Theorem 4(Birkhoff’s Erogodic Theorem). *Let*(*X,B,T,μ*)*be an ergodic system and f∈L*^{1}(*μ*)*. Then*
*for almost all x,*

*n*lim*→*∞

1
*n*

*n**−*1
*i*

### ∑

=0*f*(*T*^{i}*x*) = 1
*μ*(*X*)

*f dμ.*
Now we can prove Theorem of Lévy.

*Proof of Theorem of Lévy.* Recall that the Gauss map*T* is a shift map on continued fraction,
*T*[0;*a*1*,a*2*,...*] = [0;*a*2*,a*3*,...*]*.*

Hence for*x*= [0;*a*1*,a*2*,...*]with*a**i**≥*1,

*T*^{k}*x*= [0;*a**k*+1*,a**k*+2*,...*]*.*

So we can write

*x*= *p**n**−*1*T*^{n}*x*+*p**n*

*q**n**−*1*T*^{n}*x*+*q**n*

(8) and

*T*^{n}*x*=*−* *q**n**x−p**n*

*q**n**−*1*x−p**n**−*1*.* (9)

By plugging (9) into the numerator of (8), we have

*x*= *−p**n**−*1*q**n**x*+*p**n**q**n**−*1*x*
(*q**n**−*1*T*^{n}*x*+*q**n*)(*q**n**−*1*x*+*q**n*)

= (*−*1)^{n}^{−}^{1}
*q**n**−*1*T*^{n}*x*+*q**n*

1
*q**n**−*1*x*+*q**n**,*
which implies

*q**n**|q**n**−*1*x−p**n**−*1*|*= *q**n*

*q**n**−*1*T*^{n}*x*+*q**n**.*
Since

1*>* *q**n*

*q**n**−*1*T*^{n}*x*+*q**n* *>* *q**n*

2*q**n* =1
2*,*
we have

1*>q**n**|q**n**−*1*x−p**n**−*1*|>*1
2*.*
From (9), we can deduce that

*xT xT*^{2}*x...T*^{n}*x*= (*−*1)^{n}^{+}^{1}(*p**n**x−q**n*) =*|q**n**x−q**n**|,* (10)

that is,

1*>q**n**·xT xT*^{2}*x...T*^{n}*x>* 1
2*.*
Taking log on each side of the inequalities, we have

*|*log*q**n*+^{n}

### ∑

^{−}

^{1}

*i*=0

log*T*^{i}*x|<*log 2 (11)
Applying Birkhoff’s Ergodic Theorem on (11), we can deduce that

*n*lim*→*∞

log*q**n*

*n* =*−*lim

*n**→*∞

1
*n*

*n**−*1
*i*

### ∑

=0log(*T*^{i}*x*)

=*−*^{} ^{1}

0

log*xdμ*(*x*)

= *π*^{2}

12 log 2*,*which will be denoted by*β,*

(12)

proving Theorem of Lévy.

Combining the concept of Lévy constants, C. Faivre proved the following results (see Theorem 2 of [6]):

Theorem 5. *If x has a Lévy constantβ*(*x*)*and its partial quotient a**n*(*x*)*satisﬁes a**n*(*x*) =*o*(*α*^{n})*for all*
*α>*1*, then we have*

*n*lim*→*∞

*k**n*(*x*)

*n* =log 10
2*β*(*x*)*.*
Note that Theorem of Lochs can be induced from Theorem 5.

The equality (10) has another application on showing the result about the convergence of continued fraction expansion of an irrational number.

Proposition 4. *For each n≥*1*and x∈*[0*,*1]*, we have|*log*x−*log_{q}^{p}^{n}

*n* *|≤* _{2}*n−*2^{1} *.*
*Proof.* We split the case into two:

Case 1:*n*is even For even*n*, we have the following from Proposition 2 and (10):

• *p**n**−*1*q**n**−p**n**q**n**−*1=1

• *−*(*q**n**−*1*x−p**n**−*1) =_{q}_{n−}_{1}_{T}^{1}*n**x*+*q*_{n}

From (8)

*x*= *p**n**−*1*T*^{n}*x*+*p**n*

*q**n**−*1*T*^{n}*x*+*q**n*

= *p**n**−*1*T*^{n}*x*

*q**n**−*1*T*^{n}*x*+*q**n*+ *p**n*

*q**n**−*1*T*^{n}*x*+*q**n*

= *p**n**−*1*T*^{n}*x*

*q**n**−*1*T*^{n}*x*+*q**n**−p**n*(*q**n**−*1*x−p**n**−*1)*.*

Thus

(1+*p**n**q**n**−*1)*x*=*p**n**−*1

*T*^{n}*x*

*q**n**−*1*T*^{n}*x*+*q**n*+*p**n*

*p**n**−*1*q**n**x*=*p**n**−*1

*T*^{n}*x*

*q**n**−*1*T*^{n}*x*+*q**n*+*p**n*

*q**n*

*p**n*

*x*= *T*^{n}*x*

*p**n*(*q**n**−*1*T*^{n}*x*+*q**n*)+1*.*
Taking log on both sides, we obtain

log*x−*log*p**n*

*q**n* =log

1+ *T*^{n}*x*

*p**n*(*q**n**−*1*T*^{n}*x*+*q**n*)

*.*
By the inequality log(1+*x*)*≤x*for*x>−*1,

log*x−*log*p**n*

*q**n* *≤* *T*^{n}*x*

*p**n*(*q**n**−*1*T*^{n}*x*+*q**n*)*≤* 1

*p**n**q**n* *≤* 1
2^{n}^{−}^{2}*.*
The last inequality comes from the elementary property of continued fractions,

*p**n*=*a**n**p**n**−*1+*p**n**−*2*≥*2*p**n**−*2*≥*2^{2}*p**n**−*4*≥ ··· ≥*2^{n−}^{2}^{1}
and

*q**n*=*a**n**q**n**−*1+*q**n**−*2*≥*2*q**n**−*2*≥*2^{2}*q**n**−*4*≥ ··· ≥*2^{n−}^{2}^{1}*,*
so that*p**n**q**n**≥*2^{n}^{−}^{2}.

Case 2:*n*is odd For odd*n*, we have:

• *p**n**−*1*q**n**−p**n**q**n**−*1=*−*1

• *q**n**−*1*x−p**n**−*1=_{q}_{n−}_{1}_{T}^{1}*n**x*+*q*_{n}.
Thus

*x*= *p**n**−*1*T*^{n}*x*

*q**n**−*1*T*^{n}*x*+*q**n*+*p**n**q**n**−*1*x−p**n**q**n*

(1*−p**n**q**n**−*1)*x*=*p**n**−*1*p**n*

*T*^{n}*x*

*p**n*(*q**n**−*1*T*^{n}*x*+*q*_{n})*−*1

*−p**n**−*1*q**n**x*=*p**n**−*1*p**n*

*T*^{n}*x*

*p**n*(*q**n**−*1*T*^{n}*x*+*q*_{n})*−*1

*.*
Taking log on both sides,

log*x−*log*p**n*

*q**n* =log

1*−* *T*^{n}*x*

*p**n*(*q**n**−*1*T*^{n}*x*+*q**n*)

*.*
By the inequality log(1*−x*)*≥x*for*x>*0,

log*x−*log*p**n*

*q**n* *≥* *T*^{n}*x*
*p**n*(*q**n**−*1*T*^{n}*x*+*q**n*)
log*p**n*

*q**n* *−*log*x≤ −* *T*^{n}*x*

*p**n*(*q**n**−*1*T*^{n}*x*+*q**n*) *≤* *T*^{n}*x*

*p**n*(*q**n**−*1*T*^{n}*x*+*q**n*) *≤* 1
2^{n}^{−}^{2}*.*
Thus we have shown that for every*n≥*1,*|*log*x−*log_{q}^{p}^{n}

*n* *|≤* _{2}*n−*^{1}2.
This proposition will be useful in later section.

2.3 A Central Limit Theorem for the sequence*k*_{n}(*x*)

C. Faivre proved that a central limit theorem is valid for the sequence (*k**n*(*x*)) in [7] by using Theorem
of Lochs, Theorem of Lévy, and a dynamical results of digits of continued fraction.

Note that we can see the digits of continued fraction expansion of an irrational number as a sequence
(*a**n*(*x*))*n**≥*0. Considering each digit*a**n*(*x*)as a random variable, the sequence (*a**n*(*x*))*n**≥*0is a sequence
of random variables, in other words, a stochastic process. In particular,(*a**n*(*x*))_{n}_{≥}_{0}also satisﬁes a good
conditions, called a stationary process with uniform-mixing property (see p. 166 of [10]). See the below
deﬁnitions.

Deﬁnition 2(Uniform-Mixing). *A stationary process*(*X**n*)*n**≥*1 *is called uniform-mixing orφ-mixing if*
*there exist a sequence*(*φ**n*)_{n}_{≥}_{1}*such tat*

*–* *φ**n**→*0*as n→*∞ *and*

*– for all A∈σ*(*X*1*,X*2*,···,X**t*)*, B∈σ*(*X**t*+*n**,···*)*and n≥*1*,|P*(*A∩B*)*−P*(*A*)*P*(*B*)*|≤φ**n**P*(*A*)
*whereσ*(*X*1*,X*2*,···,X**t*)*is theσ-algebra generated by X*1*,X*2*,···,X**t**.*

More precisely, the stationary process (*a**n*) satisﬁes more powerful mixing inequality (see p. 169
of [9])

*|P*(*A∩B*)*−P*(*A*)*P*(*B*)*|≤Cr*^{n}*P*(*A*)*P*(*B*)*,*

for some positive constants*C*and 0*<r<*1. Such uniform-mixing stationary process satisﬁes following
theorem, which is proved in pp. 187-190 in [10].

Theorem 6. *Let*(*X**n*)_{n}_{≥}_{1}*be aφ-mixing stationary process with*∑*n**≥*1

*√φ**n**<*∞*. Consider a stationary*
*process*(*Y**n*)*n**≥*1*satisfying the following:*

*(i).* E(*Y**n*)*= 0 and Var(Y**n**) <*∞

*(ii). Y**n**is of the from Y**n*= *f*(*X**n**,X**n*+1*,···*)

*(iii).* ∑*n**≥*1*β**n**<*∞*, whereβ**n*=*Y*1*−*E(*Y*1*|X*1*,X*2*,···,X**n*)_{2}*.*
*Let S**n*=*Y*1+*Y*2+*···*+*Y**n**for all n≥*1*. Then for anyλ>0,*

lim sup

*n**→*∞ *P*

1max*≤**i**≤**n**|S**i**|≥λ√*
*n*

*≤*16*K*
*λ*^{2}
*with K*=sup_{n}_{≥}_{1}^{E(}^{S}_{n}^{n}^{2}^{)}*<*∞*.*

Applying Theorem 6 to the uniform-mixing stationary process(*a**n*(*x*))of digits of continued fraction
expansions of*x*, C. Faivre proved the following central limit theorem for*k**n*(*x*):

Theorem 7. *For any real x,*

*n*lim*→*∞*m{x∈*[0*,*1]:*k**n*(*x*)*−αn*
*σ√*

*n* *≤z}*= 1

*√*2*π*

_{z}

*−*∞exp(*−t*^{2}*/*2)*dt,* (13)
*for some constantσ* *>*0*,α* =6 log 2 log 10

*π*^{2} *, and m is the Lebesgue measure.*

2.4 Transfer Operators and Measure of Exceptional Set
For*ε>*0, consider a set

*E*(*ε*) =*{|k**n*(*x*)

*n* *−α* *|≥ε},*
where*α*= 6 log 2 log 10

*π*^{2} . Note that Theorem of Lochs says that the set *E*(*ε*)has measure zero for almost
all *ε* and*x*, hence we call it an exceptional set in this thesis. The last theorem is given by C. Faivre,
providing an upper bound of the measure of the exceptional set for any*ε>*0.

Theorem 8. *For allε* *>*0*, there exist constants C>*0*and*0*<λ* *<*1*such that*
*P*(*|k**n*(*x*)

*n* *−α|≥ε*)*≤Cλ*^{logn}
*for any integer N.*

The theorem will be proved by showing the followings:

lim sup

*n**→*∞

1
*n*log*P*

*k**n*(*x*)

*n* *≤α−ε*

*≤θ*1(*ε*) (0*<ε* *<α*)
and

lim sup

*n**→*∞

1
*n*log*P*

*k**n*(*x*)

*n* *≥α*+*ε*

*≤θ*2(*ε*)
where*θ*1and*θ*2are deﬁned by

*θ*1(*ε*) = inf

0*<**t**<*1*/*2

1

*t*+1(*−t*+ (*α−ε*)log*λ*(2*−*2*t*))*<*0
and

*θ*2(*ε*) = inf

*η>*0(*η*+ (*α*+*ε*)log*λ*(2+2*η*))*<*0*.*

Note that we can derive Theorem of Lochs as a corollary of this theorem with Borel-Cantelli lemma
(see [11]), saying that this result is more powerful than Theorem of Lochs. The upper bound*λ* in the
statement is the dominant eigenvalue of a transfer operator. The appearance of eigenvalue of trans-
fer operator comes from the following proposition, which is about its connection with the measure of
exceptional set.

Proposition 5. *(i). For each a>*0*, there exists a constant C such that*
E

1
*q**n*2*a*

*≤Cλ*^{n}(2+2*a*) (14)

*(ii). For each t<*^{1}_{2}*, there exist a constant C such that*

E(*q**n*2*t*)*≤Cλ*^{n}(2*−*2*t*)*.* (15)
*.*