Number theory/Signed Articles/Elementary diophantine approximations: Difference between revisions
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Fraction <math>\frac{a}{c}</math> is called the '''left neighbor''' of the other, and <math>\frac{b}{d}</math> is called the '''right neighbor'''. | Fraction <math>\frac{a}{c}</math> is called the '''left neighbor''' of the other, and <math>\frac{b}{d}</math> is called the '''right neighbor'''. | ||
* Let <math>\ a,b,c,d\in\mathbb{N}.</math> Fractions <math>\frac{a}{c}</math> and <math>\frac{b}{d}</math> are neighbors <math>\Leftrightarrow</math> fractions <math>\frac{d}{b}</math> and <math>\frac{c}{a}</math> are neighbors. | |||
'''Examples:''' | '''Examples:''' |
Revision as of 20:10, 16 January 2008
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
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 non-negative numerators and positive denominators, are called neighbors (in the given order)
Fraction is called the left neighbor of the other, and is called the right neighbor.
- 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 1 Let fractions and with positive integer numerators and denominators, be neighbors. Then
- if positive integers and are such that then
- the median is a right neighbor of and a left 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, let
Then or and the rest is obvious.
End of proof
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 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 \in \mathbb{Z}} are such that two of them are relatively prime and 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,} then any two of them are relatively prime.
- Corollary If 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} and 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 \ b} are relatively prime then also 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} and 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\,|} are relatively prime.
Now, let's define inductively a table odd integers:
- 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 (\nu_{k,n} : k\in \mathbb{Z}_+,\ 0\le n\le 2^k)}
as follows:
- 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 \nu_{0,0} := 0\ } and 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 \nu_{0,1} :=1\ }
- 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 \nu_{k+1,2\cdot n}\ :=\ \nu_{k,n}} for 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 0\le n\le 2^k\ }
- 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 \nu_{k+1,2\cdot n+1}\ :=\ \nu_{k,n}+\nu_{k,n+1}} for 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 0\le n < 2^k\ }
for every 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 \ k=0,1,\dots.}
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, 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 \ \nu_{k,n}} and 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 \ \nu_{k,n+1}} is relatively prime.
- For every pair of relatively prime, non-negative integers 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} and 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 \ b} there exist indices 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 k\in\mathbb{Z}_+} and non-negative 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 \ n<2^k} such that:
- 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\}\ =\ \{\nu_{k,n},\nu_{k,n+1}\}}
Proof Of course the pair
- 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 \{\nu_{0,0},\nu_{0,1}\}\ =\ \{0,1\}}
is relatively prime; and the inductive proof of the first statement of Theorem 2 is now instant thanks to Theorem 1 above.
Now let 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} and 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 \ b} be a pair of relatively prime, non-negative integers. If 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=1} then 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\}=\{0,1\}=\{\nu_{0,0},\nu_{0,1}\},} and the second part of the theorem holds. Continuing this unductive proof, let's assume that 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>1.} Then 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 \ \min(a,b) > 0.} Thus
But integers 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 \ c := \min(a,b)} and 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 \ d := |a-b|} are relatively prime (see Corollary above), and
- 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 c+d\ =\ max(a,b)\ <\ a+b}
hence, by induction,
- 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 \{c,d\}\ =\ \{\nu_{k,n},\nu_{k,n+1}\}}
for certain indices and non-negative 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 \ n<2^k.} Furthermore:
- 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\}\ =\ \{\min(a,b), \max(a,b)\}}
It follows that one of the two options holds:
- 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\}\ =\ \{\nu_{k+1,2\cdot n},\nu_{k+1,2\cdot n+1}\}}
or
- 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\}\ =\ \{\nu_{k+1,2\cdot n+1},\nu_{k+1,2\cdot n+2}\}}
End of proof
Let's note also, that
- 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 \max_{\quad 0\le n \le 2^k}\nu_{k,n}\ =\ F_{k+1}}
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 \ F_r}
is the r-th Fibonacci number.
Matrix monoid 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 \mathit{SO}(\mathbb{Z}_+, 2)}
Definition 1 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 \mathit{SO}(\mathbb{Z}_+, 2)} is the set of all matrices
- 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 M\ :=\ \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]}
such that 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}_+,} 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 \ \det(M) := a\cdot d - b\cdot c.} Such matrices (and their columns and rows) will be called special.
- If
- 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 M\ :=\ \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]\in\mathit{SO}(\mathbb{Z}_+, 2)}
then 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,d > 0,} and each of the columns and rows of M, i.e. each of the four pairs 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 (x,y) \in \{a,d\}\times \{b,c\},\ } is relatively prime.
Obviously, the identity matrix
- 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 \mathcal{I}\ :=\ \left[\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right]}
belongs to 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 \mathit{SO}(\mathbb{Z}_+, 2).} Furthermore, 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 \mathit{SO}(\mathbb{Z}_+, 2)} is a monoid with respect to the matrix multiplication.
Example The left matrix and the right matrix are defined respectively as follows:
- and 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 \mathcal{R}\ :=\ \left[\begin{array}{cc}1 & 0\\ 1 & 1\end{array}\right]}
Obviously 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 \ \mathcal{L},\mathcal{R}\in \mathit{SO}(\mathbb{Z}_+,2).} 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:
- 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 M\cdot \mathcal{L}\ =\ \left[\begin{array}{cc}a & a+b\\ c & c+d\end{array}\right]}
and
- 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 M\cdot \mathcal{R}\ =\ \left[\begin{array}{cc}a+b & b\\ c+d & d\end{array}\right]}
Definition 2 Vectors
- 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 \left[\begin{array}{c}a\\ c\end{array}\right]} and 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 \left[\begin{array}{c}b\\ d\end{array}\right]}
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) 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 \Leftarrow:\Rightarrow} matrix formed by these vectors
- 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 M\ :=\ \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]}
belongs to 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 \mathit{SO}(\mathbb{Z}_+, 2).} Then the left (resp. right) column is called the left (resp. right) neighbor.
Rational representation
With every vector
- 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 v\ :=\ \left[\begin{array}{c}a\\ c\end{array}\right]}
such that 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 \ c\ne 0,} let's associate a rational number
- 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 \mathit{rat}(v)\ :=\ \frac{a}{c}}
Also, let
- 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 \mathit{rat}(v_{\infty})\ :=\ \infty}
for
- 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 v_{\infty}\ :=\ \left[\begin{array}{c}1\\ 0\end{array}\right]}
Furthermore, with every matrix let's associate the real closed interval (it may include the point at plus infinity)
- 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 \mathit{span}(M)\ :=\ [\mathit{rat}(w);\mathit{rat}(v)]}
and its length
- 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 |M|\ :=\ \mathit{rat}(v)-\mathit{rat}(w)}
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 \ v} is the left, and 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 \ w} 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
- 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 M\ :=\ \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]\in\mathit{SO}(\mathbb{Z}_+, 2)}
then 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 \ |M| = \frac{1}{c\cdot d}}