Table Of ContentDiophantine Approximation
Lukas Pottmeyer
January 17, 2022
ii
Preface
These lecture notes were written during the Corona-term 2020. There are surely still plenty
of typos, although already many have been found by the participants of this course. I thank
all of the participants for their interest in the topic and their hunt for typos, errors and
inaccuracies! If you notice further issues, please sent me a mail.
Many sources have been used to write these notes, and I often follow the outline of some
sections from other books quite closely. This includes:
• The presentation of the theory of continued fractions roughly follows the presentation
in [11].
• Everything on Siegel’s Lemma is essentially taken from [5].
• The proof of Dobrowolski’s theorem closely follows the outline from [8].
• the proof of Roth’s Lemma is a mixture from the proofs given in [1] and [5].
• The proof of Roth’s theorem closely follows the outline in [1].
• The proof of the Gelfond-Schneider theorem is essentially taken from [9].
• The mentioned applications of linear forms in logarithms are taken from [3].
iii
Notations
Z the integers
Q the rational numbers
R the real numbers
C the complex numbers
N the positive integers {1,2,3,...}
N the non-negative integers {0,1,2,3,...}
0
Q a fixed algebraic closure of Q contained in C
bαc the Gauß-bracket of α (largest integer smaller or equal to the real number α)
{α} the fractional part of α (α−bαc)
|z| the usual absolute value of z ∈ C
|M| the cardinality of the set M
M(α) the Mahler measure of α ∈ Q
K the completion of the number field K with respect to the absolute value v on K
v
d the local degree [K : Q ]
v v p
set of pairwise non equivalent non-trivial absolute values v on the number field
M K, normalized such that the restriction of an archimedean v to Q is the usual |.|,
K
and the restriction of a non-archimedean v to Q is a usual p-adic absolute value
O the ring of integers of a number field K
K
H(α) the absolute multiplicative Weil-height of α ∈ Q
h(α) the absolute logarithmic Weil-height of α ∈ Q
iv
Contents
1 Foundations 1
1.1 Introduction to Diophantine Approximations . . . . . . . . . . . . . . . . . . 1
1.2 Finite Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Infinite Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Periodic Continued Fractions and Pell’s Equation . . . . . . . . . . . . . . . . 21
2 The Weil-Height 27
2.1 The Mahler Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Recap on Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 The Weil-height (finally) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4 Siegel’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5 On Lehmer’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.6 Siegel’s Lemma Once More . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Roth’s Theorem 73
3.1 Thue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2 Preliminaries I – Multivariable Polynomial Estimates . . . . . . . . . . . . . . 80
3.3 Preliminaries II – Linearly Independent Polynomials . . . . . . . . . . . . . . 85
3.4 Preliminaries III – Roth’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 The Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 Linear Forms in Logarithms 115
4.1 The Gelfond-Schneider Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
v
vi CONTENTS
Chapter 1
Foundations
1.1 Introduction to Diophantine Approximations
The word Diophantine usually refers to the integers or rationals. Hence, the main goal in
the field of Diophantine Approximation is to approximate real numbers by rational num-
bers. Since Q is dense in R, this is always possible to an arbitray accuracy. However, some
approximations are nicer than others. For instance1
314159
π = 3,1415926... ≈ (“good approximation with huge denominator”)
100000
355
≈ (“better approximation with small denominator”)
113
So, in some intuitive way, the latter approximation is much nicer.
Formally, for a given α ∈ R, we want to study p ∈ Q, with p ∈ Z, q ∈ N, and gcd(p,q) = 1,
q
such that |α−p| is small. This is surely the case if |qα−p| is small. Hence, we will frequently
q
work with this latter quantity.
Definition 1.1.1. For α ∈ R we define the Gauß-bracket of α as the largest integer bαc
smaller or equal to α. The fractional part of α is {α} = α−bαc.
Remark 1.1.2. The function b·c : R −→ Z is also known as the floor function. To be able to
distinguish between the fractional part of α and the set with the only element α (although it
should always be clear from the context), we put the brackets of the fractional part in bold
face.
Note that we always have {α} ∈ [0,1) for all α ∈ R.
Example 1.1.3. It is
• bπc = 3 and {π} = 0,1415926...
j k
• 3 = 0 and {3} = 3
4 4 4
• b−ec = b−2,7182...c = −3 and {−e} = −e+3 = 0,2817...
1the first few digits of π can be remembered by: How I want a drink? Alcoholic of course! (just count the
letters of each word)
1
2 CHAPTER 1. FOUNDATIONS
Lemma 1.1.4. Let α = a ∈ Q, with a ∈ Z, b ∈ N, and gcd(a,b) = 1. Let p,q ∈ Z such that
b
α 6= p, then |qα−p| ≥ 1.
q b
Proof. This is trivial: We have |qa−pb| = 0 if and only if α = a = p. If this is not the case,
b q
we have
(cid:12) a (cid:12) |qa−pb| 1
(cid:12) (cid:12)
|qα−p| = (cid:12)q −p(cid:12) ≥ ≥ .
(cid:12) b (cid:12) b b
Notation 1.1.5. Usually, when speaking of a rational number p, we mean that p ∈ Z, q ∈ N,
q
and gcd(p,q) = 1.
Theorem 1.1.6 (Dirichlet). Let α ∈ R be arbitrary. For each Q ∈ N, there are coprime
p,q ∈ Z, with q ∈ {1,...,Q}, such that
1
|qα−p| < . (1.1)
Q
Proof. The following proof is most beautiful! We cut the interval [0,1) into Q subintervals of
equal size; i.e.
1 1 2 Q−1
[0,1) = [0, )∪[ , )∪...∪[ ,1).
Q Q Q Q
| {z } | {z } | {z }
=:I1 =:I2 =:IQ
The Q+1 numbers {0·α}, {1·α}, ..., {Q·α} all lie in [0,1). Now, we have Q+1 numbers
in Q subintervals, and it follows that two of these numbers must lie in the same subinterval.
Say {a·α},{b·α} ∈ I , for a,b,∈ {1,...,Q}, k ∈ {1,...,Q}, and a > b. Then
k
1
> |{a·α}−{b·α}| = |a·α−b·α−ba·αc+bb·αc|
Q
= |(a−b)·α−(ba·αc−bb·αc)|.
Hence, (1.1) is fulfilled with q = a−b ∈ {1,...,Q} and p = ba·αc−bb·αc ∈ Z.
Corollary 1.1.7. A real number α ∈ R is irrational (i.e. ∈ R\Q) if and only if there are
sequences (pn)n∈N ∈ Z and (qn)n∈N ∈ N, such that
0 6= |q α−p | −→ 0 as n tends to infinity. (1.2)
n n
Proof. If α is irrational, then surely |qα−p| =6 0 for all p ∈ Z and q ∈ N. By Theorem 1.1.6,
for any n ∈ N, we can find p ∈ Z and q ∈ N, such that |q α−p | < 1. Hence, (1.2) follows.
n n n n n
If on the other hand α is rational, then (1.2) is not satisfied by Lemma 1.1.4.
Slogan: Irrational numbers can be better approximated than rational numbers!
Example 1.1.8. We give two applications:
j√ k (cid:12)√ (cid:12)
(a) We have 2 = 1 (since 12 < 2 and 22 > 2). Hence, (cid:12) 2−1(cid:12) < 1. It follows that
(cid:12) (cid:12)
(cid:12) √ (cid:12) √ √
0 6= (cid:12)( 2−1)n(cid:12) tends to zero as n tends to infinity. Since ( 2−1)n ∈ Z[ 2], for any
(cid:12) (cid:12)
√ √
n ∈ N, there are p ,q ∈ Z such that ( 2−1)n = q 2−p . This means that (1.2) is
n n √ n n
satisfied, and therefore 2 is irrational. We all knew this before, but this proof does not
need any knowledge of prime decomposition!
If you like to have it slightly more explicit, one can show that
1.1. INTRODUCTION TO DIOPHANTINE APPROXIMATIONS 3
• p =−1, p =1, p =−3, and p =−2p +p for all n≥2
0 1 2 n+1 n n−1
• q =0, q =1, p =−2, and q =−2q +q for all n≥2.
0 1 2 n+1 n n−1
(b) Euler’s constant e equals P∞ 1. Set q = n! ∈ N and p = n!·Pn 1 ∈ N. Then, it is
i=0 i! n n i=0 i!
X∞ 1
0 6= |q e−p | = n!·
n n
(n+i)!
i=1
X∞ 1 X∞ 1
= <
(n+1)(n+2)·...·(n+i) (n+1)i
i=1 i=1
1 1
n→∞
= −1 = −→ 0.
1− 1 n
n+1
By Corollary 1.1.7 it follows that e is irrational.
The next corollary to Theorem 1.1.6 is the first major result of Diophantine Approximation,
that we come across.
Corollary 1.1.9. Let α ∈ R be irrational. There are infinitely many rational numbers p,
q
with gcd(p,q) = 1, q ≥ 1, such that
(cid:12) p(cid:12) 1
(cid:12) (cid:12)
(cid:12)α− (cid:12) < . (1.3)
(cid:12) q(cid:12) q2
Proof. Actually, the proof is almost immediately clear from Theorem 1.1.6. But as the result
is that important, we will give the proof in full detail.
Assume that for some irrational α there are only finitely many – say precisely n – rational
(cid:12) (cid:12)
numbers p1,..., pn satisfying (1.3). Since α ∈/ Q, we have 0 6= (cid:12)α− pi(cid:12) for all i ∈ {1,...,n}.
q1 qn (cid:12) qi(cid:12)
(cid:12) (cid:12)
Take any Q ∈ N satisfying 1 < min (cid:12)α− pi(cid:12). By Theorem 1.1.6, there are coprime
Q i∈{1,...,n}(cid:12) qi(cid:12)
p,q ∈ Z, with q ∈ {1,...,Q}, such that
(cid:12) p(cid:12) 1 1 (cid:12) p (cid:12)
(cid:12) (cid:12) (cid:12) i(cid:12)
0 6= (cid:12)α− (cid:12) = ·|qα−p| < < (cid:12)α− (cid:12) ∀ i ∈ {1,...,n}.
(cid:12) q(cid:12) q qQ (cid:12) q (cid:12)
i
In particular, p 6= pi for all i ∈ {1,...,n}. Moreover, as q ∈ {1,...,Q}, we have
q qi
(cid:12) p(cid:12) 1 1 1
(cid:12) (cid:12)
(cid:12)α− (cid:12) = ·|qα−p| < ≤ .
(cid:12) q(cid:12) q qQ q2
So,thereareatleastn+1rationalnumberssatisfying(1.3),whichcontradictsourassumption.
Hence, there must be infinitely many rational numbers satisfying (1.3). This proves the
corollary.
Remark 1.1.10. Theexponent2in(1.3)willkeepusbusyforsometime. Anaturalquestion
is, whether the statement of Corollary 1.1.9 remains true if 2 is replaced by 3, or 4, or 5, ...
Note that increasing the exponent makes the approximations better!
Notation 1.1.11. Throughout the lectures, we will fix an algebraic closure of Q in C and
denote it by Q. An element in α ∈ Q is called algebraic or an algebraic number. This means,
that there are integers a ,...,a , with a 6= 0, such that a αd +a αd−1 +...+a = 0.
0 d d d d−1 0
4 CHAPTER 1. FOUNDATIONS
A different way to formulate this, is that there is a polynomial f(x) ∈ Z[x] \ {0}, with
f(α) = 0. In fact, there is a unique (up to multiplication by −1) irreducible polynomial
f(x) ∈ Z[x] satisfying f(α) = 0. Such a polynomial is called minimal polynomial of α. If
f(x) = a xd+a xd−1+...+a istheminimalpolynomialofα,thenk = gcd(a ,...,a ) = 1,
d d−1 d d 0
since otherwise f(x) = k·(adxd+...+ a0) would not be irreducible.
k k
An equivalent definition for an algebraic number is, that α ∈ C is algebraic if and only if
[Q(α) : Q] is finite. In this case, [Q(α) : Q] equals the degree of the minimal polynomial of α,
and is called the degree of α.
You have learned all of this in your Algebra class, to which we refer for further details!
Theorem 1.1.12 (Liouville). Let α ∈ Q be of degree d ≥ 2 (i.e. an algebraic irrational
number). Then there is a constant c(α) > 0 such that for all rational numbers p, q ∈ N, we
q
(cid:12) (cid:12)
have (cid:12)α− p(cid:12) > c(α).
(cid:12) q(cid:12) qd
Proof. We will present an analytic proof here. Later we will present an algebraic proof as
well.
Let f(x) = a xd+a xd−1+...+a be the minimal polynomial of α. Since d ≥ 2, there is
d d−1 0
a non-constant polynomial g(x) ∈ C[x] such that f(x) = (x−α)·g(x). Since any irreducible
polynomial over Z[x] is separable (does not have multiple roots), we know that g(α) 6= 0.
Also, since f(x) is irreducible, it has no roots in Q, and hence g(p) 6= 0. Moreover, as a
q
polynomial, g(x) is a continuous self-map on C. Let ε > 0 be smaller than the distance of α
to any other root of f(x). Then, g(β) 6= 0 for all β ∈ C, with |α−β| < ε. So in particular
0 6= δ = sup |g(β)|. Since g(x) is continuous on C, we also know that δ 6= ∞. Hence,
|α−β|<ε
c(α) := min{ε,δ−1} ∈ R.
(cid:12) (cid:12)
Assume there are p ∈ Z, q ∈ N, such that (cid:12)α− p(cid:12) ≤ c(α). Then
(cid:12) q(cid:12) qd
(cid:12) p(cid:12) (cid:12) p (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)α− (cid:12) < ε =⇒ 0 6= (cid:12)g( )(cid:12) < δ
(cid:12) q(cid:12) (cid:12) q (cid:12)
(cid:12) p (cid:12) (cid:12) p(cid:12) (cid:12) p (cid:12) c(α) 1
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
=⇒ (cid:12)f( )(cid:12) = (cid:12)α− (cid:12)·(cid:12)g( )(cid:12) < ·δ ≤ .
(cid:12) q (cid:12) (cid:12) q(cid:12) (cid:12) q (cid:12) qd qd
Multiplying both sides of the latter inequality by qd yields
(cid:12) p (cid:12)
1 > (cid:12)(cid:12)qd·f( )(cid:12)(cid:12)
(cid:12) q (cid:12)
= (cid:12)(cid:12)(cid:12)(cid:12)qd·(ad(pq)d+ad−1(pq)d−1+...+a0(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)adpd+ad−1pd−1q+...+a0qd(cid:12)(cid:12)(cid:12) ∈ N0.
(cid:12) (cid:12)
It follows that (cid:12)qd·f(p)(cid:12) = 0, which contradicts f(p) 6= 0.
(cid:12) q (cid:12) q
Remark 1.1.13. Theorem 1.1.12 tells us, that algebraic numbers cannot have “arbitrarily
good” approximations. So maybe we can use this to prove the transcendence of certain
complex numbers. Recall that an α ∈ C is called transcendental over Q if it is not algebraic.
Cantor provided a remarkable argument for the existence of transcendental numbers: The
set of complex numbers is uncountable by Cantor’s diagonal argument. On the other hand
Q consists of roots of polynomials in Z[x]. We have
∞
Z[x] = [ {a +a x+...+a xd|d ∈ N,a ,...,a ∈ Z, and d+|a |+...+|a | = N}.
0 1 d 0 n 0 d
N=0
