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

  •   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 ";
  •  —  the greatest common divisor of integers   and

 

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



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 left matrix and the right 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a,b,c,d\in\mathbb{Z}_+,}   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