Number theory/Signed Articles/Elementary diophantine approximations: Difference between revisions
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where in each of these two cases <math>\ n\in \mathbb{N}</math> is a respective unique positive integer. | where in each of these two cases <math>\ n\in \mathbb{N}</math> is a respective unique positive integer. | ||
It was mentioned in the previous section ('''''First results''''') that if fractions <math>\frac{a}{c}</math> and <math>\frac{b}{d},</math> with positive (or non-negative) integer numerators and denominators are neighbors then also the top and the bottom (''bot'' for short) pair: | It was mentioned in the previous section ('''''First results''''') that if fractions <math>\frac{a}{c}</math> and <math>\frac{b}{d},</math> with positive (or non-negative) integer numerators and denominators are neighbors then also the top and the bottom (''bot'' for short) pair: | ||
*<math>\mathit{top}\!\left(\frac{a}{c},\frac{b}{d}\right)\ :=\ \left(\frac{a}{c},\frac{a+b}{b+d}\right)</math> | *<math>\mathit{top}\!\left(\frac{a}{c},\frac{b}{d}\right)\ :=\ \left(\frac{a}{c},\frac{a+b}{b+d}\right)</math> |
Revision as of 06:40, 19 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
- (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, and is called the bottom neighbor.
- 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 positive integer numerators and denominators, be neighbors. Then
- if positive 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 Since
it is enough to prove Hurwitz theorem for positive irrational numbers only. Thus from now on let Consider the sequence of pairs of neighbors, which squeeze from the previous section. The case of the infinite alternation top-bot-top-... has been proved already. In the remaining case ... (to be continued)
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 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 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.} 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 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) = 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 \ \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 upper matrix and the lower matrix are defined respectively 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 \mathcal{U}\ :=\ \left[\begin{array}{cc}1 & 1\\ 0 & 1\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 \mathcal{L}\ :=\ \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{U},\mathcal{L}\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{U}\ =\ \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{L}\ =\ \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
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 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\in \mathit{SO}(\mathbb{Z}_+,2),} let's associate the real open interval
- 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}}