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.
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