Number theory/Signed Articles/Elementary diophantine approximations

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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 if and only if there exists a real number C > 0 such that

for arbitrary integers   such that   and

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

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 (for the sake of simplicity only positive numbers will be considered). 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.

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.

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:

 

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, either

or

and we get the respective required inequality in each case:

or, respectively,

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 — 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

(apply theorem from section First results). Let's get a sharper result:

First of all, since is irrational, the direction top-bottom of the sequence of pairs of neighbors changes infinitely many times, i.e.


for infinitely many values of     and


for infinitely many values of    as well. Thus there are infinitely many different   of each of the two kinds. The other two kinds (which may or may not actually occur) are described by conditions:

and


Let   be a pair of neighbors for which the top-top property above holds, i.e. for which   is squeezed between a pair of neighbors as follows:

Then


Similarly, in the bot-bot case we get


Thus if the top-top or the bot-bot case holds infinitely many times then there exist infinitely many fractions of natural numbers    such that:

On the other hand, 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. It is known that

hence

Thus

and

If    then

and if    i.e.    then

Since  and due to the earlier inequalities which have covered the top-top and the bot-bot cases, we have obtained the following theorem:

  • for arbitrary irrational number there exist infinitely many different fractions   with integer numerator and non-zero denominator, such that:

However, we are close to replacing constant    by    in the above denominator. Let's do it:

 

Hurwitz theorem

  • Let    be an arbitrary irrational number. Then

for infinitely many different


Proof of Hurwitz Theorem

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



Let    Then



 

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