Number theory/Signed Articles/Elementary diophantine approximations

The theory of diophantine approximations is a chapter of number theory, which in turn is a part of mathematics. It studies the approximations of real numbers by rational numbers. This article presents an elementary introduction to diophantine approximations, as well as an introduction to number theory via diophantine approximations.

Introduction

In the everyday life our civilization applies mostly (finite) decimal fractions   Decimal fractions are used both as certain values, e.g. \$5.85, and as approximations of the real numbers, e.g.   However, the field of all rational numbers is much richer than the ring of the decimal fractions (or of the binary fractions   which are used in the computer science). For instance, the famous approximation   has denominator 113 much smaller than 105 but it provides a better approximation than the decimal one, which has five digits after the decimal point.

How well can real numbers (all of them or the special ones) be approximated by rational numbers? A typical Diophantine approximation result states:

Theorem  Let   be an arbitrary real number. Then

•   is rational      there exists a real number C > 0 such that


for arbitrary integers   such that   and 

• (Adolph Hurwitz)     is irrational      there exist infinitely many pairs of integers   such that   and


Remark  Implication      of the first part of the theorem is a simple and satisfaction bringing exercise.

Notation

•    —   "equivalent by definition" (i.e. "if and only if");
•    —   "equals by definition";
•    —   "there exists";
•    —   "for all";
•    —   "  is an element of set ";

•   —  the semiring of the natural numbers;
•   —  the semiring of the non-negative integers;
•   —  the ring of integers;
•   —  the field of rational numbers;
•   —  the field of real numbers;

• 
• 
• 

•    —   "  divides "   (i.e. );
•   —  the greatest common divisor of integers   and 

The method of neighbors and median

In this section we will quickly obtain some results about approximating irrational numbers by rational. To this end we will not worry about the details of the difference between a rational number and a fraction (with integer numerator and denominator)—this will not cause any problems; fully crisp notions will be developed in the next sections, they will involve 2-dimensional vectors and 2x2 matrices. This section is still introductory. It is supposed to provide quick insight into the topic.

Definitions

Fractions   and   with integer numerators and natural denominators, are called neighbors (in the given order)   



Fraction   is called the top neighbor of the other,   is called the bottom neighbor, and the interval   is called neighborhood; thus a neighborhood is an open interval such that its endpoints are neighbors.

• If    and   are neighbors then     ( i.e.   ).
• Let    Fractions   and   are neighbors      fractions   and   are neighbors      fractions   and   are neighbors.

Examples:

• Fractions   and   are neighbors for every positive integer 

• Fractions   and   are neighbors for every positive integer 

Thus it easily follows that for every positive irrational number   there exists a pair of neighbors    and    with positive numerators and denominators, such that:



Definition  A pair of neighboring fractions   with integer numerators and natural denominators, is called a top pair      Otherwise it is called a bottom pair.

Thus now we have notions of top/bottom neighbors and of top/bottom pairs of neighbors.

• Let    be a pair of neighbors. Then    is a top pair of neighbors, and    is a bottom pair of neighbors.

First results

Theorem  Let fractions   and   with integer numerators and natural denominators, be neighbors. Then

• if integers   and   are such that      then   
• the median    is a bottom neighbor of    and a top neighbor of  
• let    be an irrational number such that      then

and
       or       

Proof   Let      then

     and     

and



Multiplying this inequality by   gives



which is the first part of our theorem.

The second part of the theorem is obtained by a simple calculation, straight from the definition of the neighbors.

The first inequality of the third part of the theorem is instant:



Next,



hence





and



i.e.



End of proof

Corollary  Let fractions   and   with integer numerators and natural denominators, be neighbors. Then, if integers   and   are such that      then either

• 

or

• 

Hurwitz theorem

Let    be an arbitrary irrational number. Then

for infinitely many different 

Lemma 1

Let    Let    Then:



or



Proof of lemma 1  It's easy to show that    Thus the square of    is positive. Now,



which means that we may write    as follows:



i.e.



and lemma 1 follows.  End of proof

Lemma 2

Let    and    Let    and    Furthermore, let fractions    be neighbors, and let:



where   is real. Then one of the following three inequalities holds:

• 

• 

• 

Proof  There are two cases along the inequalities of Lemma 1. Let's assume the first one, which is equivalent to:



Thus



which means that





Thus lemma 2 is proven when the first inequality of lemma 1 holds. By replacing in the above proof the lower case    by the upper case    we obtain the proof when the second inequality of lemma 1 holds.

End of proof (of lemma 2)

Lemma 2'

Let    and    Let    and    Furthermore, let fractions    be neighbors, and let:



where   is real. Then one of the following three inequalities holds:

• 

• 

• 

Proof  It's similar to the proof of lemma 2. Or one may apply lemma 2 to    and  , which would provide us with the respective fraction    Then the required    are given by  

End of proof (of lemma 2')

Proof of Hurwitz theorem

When    is irrational, then it is squeezed between infinitely many different pairs of neighbors (see part two of the theorem from the First results section). They provide infinitely many different required fractions    (see lemma 2 and 2' above; but it is not claimed, nor true, that different pairs of neighbors provide different required fractions).

End of proof

Squeezing irrational numbers between neighbors

Let   be an irrational number, We may always squeeze it between the extremal neighbours:



But if you don't like infinity (on the left above) then you may do one of the two things:



or



where in each of these two cases     is a respective unique positive integer.

It was mentioned in the previous section (First results) that if fractions   and   with positive (or non-negative) integer numerators and denominators are neighbors then also the top and the bottom (bot  for short) pair:

• 

and

• 

are both pairs of neighbors.

Let   be a pair of neighbors, and  be an irrational number. Assume that pairs of neighbors   are already defined, and that they squeeze  i.e. that   for each   Then we define   as the one of the two pairs:   or    which squeezes   Thus for every positive irrational number we have obtained an infinite sequence of pairs of neighbors, each squeezing the given irrational number more and more. Thus for arbitrary irrational   there exist fractions of integers   with arbitrarily large denominators, such that



(see section Hurwitz theorem).

If cases top-top and bot-bot happen only finitely many times then starting with an   we get an infinite alternating top-bot-top-bot-... sequence:



Then the new neighbor of the   pair (i.e. the median of the previous pair )  is equal to



for every   where   are the Fibonacci numbers, where



It is known that



hence



But if our infinite alternation has started with bot :



Then we would have



Another proof of Hurwitz Theorem (further insight)

Reduction to x > 0

Since



it is enough to prove Hurwitz theorem for positive irrational numbers only.

Two cases

Consider the sequence   of pairs of neighbors, which squeeze   from the previous section. The case of the infinite alternation top-bot-top-bot-... has been proved already. In the remaining case the bot-bot-top or top-top-bot progressions appear infinitely many times, i.e. there are infinitely many non-negative integers   for which



or



holds, where



The top-top-bot case

Let's consider the latter top-top-bot case. Let    The squeeze by neighbors:



shows that



This inequality provides the first insight (otherwise, we are not going to use it), so it deserves to be written cleanly as an implication:




The relevant neighborhoods

Consider the next two pairs of neighbors, pair   and pair   which squeeze   The relevant neighborhoods are:

• 

• 

• 

• 

Neighborhood B

Let    Then



and




Conclusion:




Neighborhood C (first C-inequality)

Let    Then



Thus using the calculation for neighborhood B also for C, we get




First C-conclusion:




Neighborhood D

Let    i.e.



Let     and     Then






Conclusion:




Early yield (Hurwitz Theorem)

Let's impatiently indulge ourselves in already getting some crude results from the above hard work (:-). The above three BCD-conclusions instantly imply:

• 

• 

• 

Since



each occurrence of the top-top-bot subsequence, i.e. of equalities:



provides a fraction    with integer numerator and natural denominator, such that



The same is holds for every occurrence of the bot-bot-top sequence, which can be shown now mechanically by a proof similar to the proof of the top-top-bot case, or as follows: define the squeezing sequence of    by:

     for     

Let    be a bot-bot-top progression. Then    is a top-top-bot progression which squeezes   Thus



for certain    with integer numerator and natural denominator. Then    for    satisfies:



When another top-top-bot or bot-bot-top progression starts with a sufficiently large index    then    which means that the respective new approximation    is different. It follows that if there are infinitely many progressions top-top-bot or bot-bot-bot then there are infinitely many fractions   which satisfy the inequality above. Thus we have obtained the following version of Hurwitz theorem:

Theorem  Let    be an arbitrary irrational number. Then inequality
• 
holds for infinitely many fractions    with integer numerator and natural denominator. Furthermore, if the squeezing sequence of   does not eventually alternate top-bot-top-bot-... till infinity, i.e. if it has infinitely many top-top or bot-bot progressions, then
• 
holds for infinitely many fractions    with integer numerator and natural denominator.

Neighborhood C (second C-inequality)

Let    Then



Thus, using the earlier conclusion for neighborhood   also for   we obtain










Second C-conclusion:




Neigborhood C (the combined inequality)

Let:



Then



Thus for  :



and for  :



It follows that

• for one of the fractions    the following inequality holds for every  



(the choice of    depends on   ).

Divisibility

Definition  Integer   is divisible by integer       

Symbolically:

      

When   is divisible by   then we also say that   is a divisor of   or that   divides 

• The only integer divisible by   is   (i.e.   is a divisor only of ).
•   is divisible by every integer.
•   is the only positive divisor of 
• Every integer is divisible by   (and by  ).

• 
• 

• 
• 
• 

Remark  The above three properties show that the relation of divisibility is a partial order in the set of natural number    and also in   is its minimal, and   is its maximal element.

• 

Relatively prime pairs of integers

Definition  Integers   and   are relatively prime        is their only common positive divisor.

• Integers   and   are relatively prime   
•   is relatively prime with every integer.
• If   and   are relatively prime then also   and    are relatively prime.

Theorem 1  If   are such that two of them are relatively prime and   then any two of them are relatively prime.
Corollary  If   and   are relatively prime then also   and    are relatively prime.

Now, let's define inductively a table odd integers:



as follows:

•      and     
•    for  
•    for  

for every  

The top of this table looks as follows:

0 1
0 1 1
0 1 1 2 1
0 1 1 2 1 3 2 3 1
0 1 1 2 1 3 1 3 1 4 3 5 2 5 3 4 1

etc.

Theorem 2
• Every pair of neighboring elements of the table,   and    is relatively prime.
• For every pair of relatively prime, non-negative integers   and   there exist indices   and non-negative   such that:


Proof  Of course the pair



is relatively prime; and the inductive proof of the first statement of Theorem 2 is now instant thanks to Theorem 1 above.

Now let   and   be a pair of relatively prime, non-negative integers. If   then   and the second part of the theorem holds. Continuing this unductive proof, let's assume that   Then   Thus



But integers   and   are relatively prime (see Corollary above), and



hence, by induction,



for certain indices   and non-negative   Furthermore:



It follows that one of the two options holds:



or



End of proof

Let's note also, that



where   is the r-th Fibonacci number.

Matrix monoid 

Definition 1    is the set of all matrices



such that   and    where    Such matrices (and their columns and rows) will be called special.

• If


then    and each of the columns and rows of M, i.e. each of the four pairs    is relatively prime.

Obviously, the identity matrix



belongs to     Furthermore,    is a monoid with respect to the matrix multiplication.

Example  The upper matrix and the lower matrix are defined respectively as follows:

   and   

Obviously   When they act on the right on a matrix M (by multipliplying M by itself), then they leave respectively the left or right column of M intact:



and



Definition 2  Vectors

     and     

where   are called neighbors (in that order)      matrix formed by these vectors



belongs to    Then the left (resp. right) column is called the left (resp. right) neighbor.

Rational representation

With every vector



such that   let's associate a rational number



Also, let



for



Furthermore, with every matrix    let's associate the real open interval



and its length



where   is the left, and   is the right column of matrix M — observe that the rational representation of the left column of a special matrix is always greater than the rational representation of the right column of the same special matrix.

• If


then