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
![{\displaystyle |a-{\frac {x}{y}}|>{\frac {C}{y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/669b355298532a783ebaab7c42a3835217876c8f)
for arbitrary integers
such that
and
is irrational if and only if there exist infinitely many pairs of integers
such that
and
![{\displaystyle |a-{\frac {x}{y}}|<{\frac {1}{{\sqrt {5}}\cdot y^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1182bc3f3b7b3186aacfa1719c7821cdb7aeb2)
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;
![{\displaystyle (x;y)\ :=\{t\in \mathbb {R} :x<t<y\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d09e7838a0ff50ddd1a6a0b00f0e4a998dcd447)
— "
divides
" (i.e.
);
— the greatest common divisor of integers
and ![{\displaystyle \ b.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3658546c70c97d710581246f2c4952e8d6997cb2)
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)
![{\displaystyle {\frac {a}{c}}-{\frac {b}{d}}\ =\ {\frac {1}{c\cdot d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6eb510c0a5f1c8e177ec9ec4f7d94437a99c6b3)
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 ![{\displaystyle \ n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91200b76008a0b9e4fcfd5aa0d156bf78b66b5b1)
- Fractions
and
are neighbors for every positive integer ![{\displaystyle \ n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91200b76008a0b9e4fcfd5aa0d156bf78b66b5b1)
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 ![{\displaystyle t\geq b+d;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c44699760a41339713105835c29b930bfd373ca6)
- the median
is a right neighbor of
and a left neighbor of
- let
be an irrational number such that
then
![{\displaystyle \max({\frac {a}{c}}-x,\,x-{\frac {b}{d}})\ <\ {\frac {1}{c\cdot d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0545903fe901c77c86730c774b3268fe69e9d14)
- and
or ![{\displaystyle x-{\frac {b}{d}}\ <\ {\frac {1}{d\cdot (d+1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f90d89e9334a90afb7ac34c1100620c4a8b8ab1)
Proof Let
then
and ![{\displaystyle {\frac {s}{t}}-{\frac {b}{d}}\ \geq \ {\frac {1}{d\cdot t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f22e2dc5c6ca1fa9f6fe5afb77ca4f75eb61bfe9)
and
![{\displaystyle {\frac {1}{c\cdot t}}+{\frac {1}{d\cdot t}}\ \leq {\frac {1}{c\cdot d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f8ff460a9af0e780e7f5660c731dc4b0a4ca37c)
Multiplying this inequality by
gives
![{\displaystyle t\geq c+d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/562ceef336c833f9bac203d81bbf99a7d78b475b)
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:
![{\displaystyle \max({\frac {a}{c}}-x,\,x-{\frac {b}{d}})\ <\ ({\frac {a}{c}}-x)+(x-{\frac {b}{d}})\ =\ {\frac {1}{c\cdot d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f055c40fac37f8c9f82e94442cfe39d6138279)
Next, either
![{\displaystyle {\frac {a}{c}}\ >\ x\ >\ {\frac {a+b}{c+d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29c92b8a365a18a48ddde8d046fc1acc27a5a1ed)
or
![{\displaystyle {\frac {a+b}{c+d}}\ >\ x\ >\ {\frac {b}{d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd66bc7486a2eddd542f01194bd3c1b6c6daf5b)
and we get the respective required inequality in each case:
![{\displaystyle {\frac {a}{c}}-x\ <\ {\frac {1}{c\cdot (c+d)}}\ \leq \ {\frac {1}{c\cdot (c+1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5b6ef66803fe86411befc460cdf7b498ecd8afe)
or, respectively,
![{\displaystyle x-{\frac {b}{d}}\ <\ {\frac {1}{d\cdot (c+d)}}\ \leq \ {\frac {1}{d\cdot (d+1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b97c07b3feafa047366364814faad02c9ef39c09)
End of proof
Squeezing irrational numbers between neighbors
Let
be an irrational number, We may always squeeze it between the extremal neighbours:
![{\displaystyle {\frac {1}{0}}\ >\ x\ >\ {\frac {0}{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0b08c1e454decd65ee8fe200a1000f9864fc9cf)
But if you don't like infinity (on the left above) then you may do one of the two things:
![{\displaystyle x>1\quad \quad \Rightarrow \quad \quad {\frac {n+1}{1}}>x>{\frac {n}{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae062497290fe64a676d26863508bf82f232ca9d)
or
![{\displaystyle x<1\quad \quad \Rightarrow \quad \quad {\frac {1}{n}}>x>{\frac {1}{n+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a92a7411d6e618549da01d1a2ecd2b5c2a966f4)
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 left and the right pair:
![{\displaystyle {\mathit {lft}}\!\left({\frac {a}{c}},{\frac {b}{d}}\right)\ :=\ \left({\frac {a}{c}},{\frac {a+b}{b+d}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44a94627e9558a3d05384e22ad7feb9283181a5e)
and
![{\displaystyle {\mathit {rgt}}\!\left({\frac {a}{c}},{\frac {b}{d}}\right)\ :=\ \left({\frac {a+b}{c+d}},{\frac {b}{d}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1477b609ff79e2050906a600ece54783c50409)
are both pairs of neighbors.
Divisibility
Definition Integer
is divisible by integer
Symbolically:
![{\displaystyle \exists _{c\in \mathbb {Z} }\ a=b\cdot c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03c81b33594cf83ba2e5b7cd4b889ade2d3c1bd3)
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 ![{\displaystyle \ 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cadcfff0f7aabf708593c25995db8a10af8f2535)
- Every integer is divisible by
(and by
).
![{\displaystyle a|b\ \Rightarrow (-a|b\ \land \ a|\!-\!\!b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e30ed473d47351c0ec679fc7510ffbaa8e1decde)
![{\displaystyle a|b>0\ \Rightarrow a\leq b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f646f9cdd4cfe3c175c6c88f7206376027e2cf2b)
![{\displaystyle a|a\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/255a27b35dd78e2626fce6914bdda12257d66807)
![{\displaystyle (a|b\ \land \ b|c)\ \Rightarrow \ a|c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c55b5c611b91641fd56fcd938b7d360823d691)
![{\displaystyle (a|b\ \land \ b|a)\ \Rightarrow \ |a|=|b|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1bd7d2f3c0c6a3ac75bdf4fa3f8ac312b04d6a6)
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.
![{\displaystyle (a|b\ \land \ a|c)\ \Rightarrow \ (a\,|\,b\!\cdot \!d\ \land \ a\,|\,b\!+\!c\ \land \ a\,|\,b\!-\!c)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/751c84610824a2406fc0e64db460198c3e72070a)
Relatively prime pairs of integers
Definition Integers
and
are relatively prime
is their only common positive divisor.
- Integers
and
are relatively prime ![{\displaystyle \Leftrightarrow \ |x|=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6063982dda364a20a30bfcce524f4ce54fda8837)
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:
![{\displaystyle (\nu _{k,n}:k\in \mathbb {Z} _{+},\ 0\leq n\leq 2^{k})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9a78813dedc946a68507b5cc1f7c2315703d629)
as follows:
and ![{\displaystyle \nu _{0,1}:=1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/63b4021472755c6b53856888a7d0ab52e915a089)
for ![{\displaystyle 0\leq n\leq 2^{k}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc256387ccca68a6686de17cb969fbef308f082b)
for ![{\displaystyle 0\leq n<2^{k}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2287f7be5314bb339a42b66ddbf060a2632322)
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:
![{\displaystyle \{a,b\}\ =\ \{\nu _{k,n},\nu _{k,n+1}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b28a3629723690a749ba0465275ff08a6a245c6)
Proof Of course the pair
![{\displaystyle \{\nu _{0,0},\nu _{0,1}\}\ =\ \{0,1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/070bcdcf7cb64763fe1196240e7cbe3c8bb890b3)
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
![{\displaystyle \ \max(a,b)<a+b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6370fe61a81edcc3fb378de582bb1fd66ba532e6)
But integers
and
are relatively prime (see Corollary above), and
![{\displaystyle c+d\ =\ max(a,b)\ <\ a+b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26e1002e93533dd91eb99a941d992ad280fb4bf2)
hence, by induction,
![{\displaystyle \{c,d\}\ =\ \{\nu _{k,n},\nu _{k,n+1}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdfcee9a88c41727a294fe6140cef17100f45e48)
for certain indices
and non-negative
Furthermore:
![{\displaystyle \{a,b\}\ =\ \{\min(a,b),\max(a,b)\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e765d368991ecb27572ded2efcb29bf041f39f88)
It follows that one of the two options holds:
![{\displaystyle \{a,b\}\ =\ \{\nu _{k+1,2\cdot n},\nu _{k+1,2\cdot n+1}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1531f97030d2cecdd7ce42e3e367b9e9f55af032)
or
![{\displaystyle \{a,b\}\ =\ \{\nu _{k+1,2\cdot n+1},\nu _{k+1,2\cdot n+2}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a671e28c40bf9ad4b426d62e011a53d84f3b6273)
End of proof
Let's note also, that
![{\displaystyle \max _{\quad 0\leq n\leq 2^{k}}\nu _{k,n}\ =\ F_{k+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed567fd5338e27b9d3247a1d3599fb2d89df56bc)
where
is the r-th Fibonacci number.
Matrix monoid ![{\displaystyle {\mathit {SO}}(\mathbb {Z} _{+},2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98d21a7d80bdccf21ae5fa7dfb8bd59cd4ca9b3c)
Definition 1
is the set of all matrices
![{\displaystyle M\ :=\ \left[{\begin{array}{cc}a&b\\c&d\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0c30ae19dc60ab9029a08ebbd73a7cf895039f)
such that
and
where
Such matrices (and their columns and rows) will be called special.
![{\displaystyle M\ :=\ \left[{\begin{array}{cc}a&b\\c&d\end{array}}\right]\in {\mathit {SO}}(\mathbb {Z} _{+},2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9746e9f95199d8ff37ecef1284d9701d6fb4e39e)
then
and each of the columns and rows of M, i.e. each of the four pairs
is relatively prime.
Obviously, the identity matrix
![{\displaystyle {\mathcal {I}}\ :=\ \left[{\begin{array}{cc}1&0\\0&1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67b0aca44b3fd67f0f11cbe7f4270cf375c119b2)
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 ![{\displaystyle {\mathcal {R}}\ :=\ \left[{\begin{array}{cc}1&0\\1&1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/724176742fcc683ca4f9bf1197205a12432d4b70)
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:
![{\displaystyle M\cdot {\mathcal {L}}\ =\ \left[{\begin{array}{cc}a&a+b\\c&c+d\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87897b41f19b3946c471f5d5d17fd38a8f8183ac)
and
![{\displaystyle M\cdot {\mathcal {R}}\ =\ \left[{\begin{array}{cc}a+b&b\\c+d&d\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5386d78c36106549c2ab6224aca1afe376be35)
Definition 2 Vectors
and ![{\displaystyle \left[{\begin{array}{c}b\\d\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0fe8c71508e4980923fe5c7f30c31aa2dc37cd)
where
are called neighbors (in that order)
matrix formed by these vectors
![{\displaystyle M\ :=\ \left[{\begin{array}{cc}a&b\\c&d\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0c30ae19dc60ab9029a08ebbd73a7cf895039f)
belongs to
Then the left (resp. right) column is called the left (resp. right) neighbor.
Rational representation
With every vector
![{\displaystyle v\ :=\ \left[{\begin{array}{c}a\\c\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8ce21cd9cf75f49607a9e5b5416ba00d707d53)
such that
let's associate a rational number
![{\displaystyle {\mathit {rat}}(v)\ :=\ {\frac {a}{c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea072b8887dbbb5ae7a034e4eb725efd0aa5961)
Also, let
![{\displaystyle {\mathit {rat}}(v_{\infty })\ :=\ \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e44c015f26f2a6516d494ceb96f687d4e4028d)
for
![{\displaystyle v_{\infty }\ :=\ \left[{\begin{array}{c}1\\0\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b87f8231efb1fd36b78f839d01924ac148aae947)
Furthermore, with every matrix
let's associate the real open interval
![{\displaystyle {\mathit {span}}(M)\ :=\ ({\mathit {rat}}(w);{\mathit {rat}}(v))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f977820210dfd95e51b2f528a5aa6ff41917e4)
and its length
![{\displaystyle |M|\ :=\ {\mathit {rat}}(v)-{\mathit {rat}}(w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/440b8958e76d42c2e34a07f64769b1f0c4e77c90)
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.
![{\displaystyle M\ :=\ \left[{\begin{array}{cc}a&b\\c&d\end{array}}\right]\in {\mathit {SO}}(\mathbb {Z} _{+},2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9746e9f95199d8ff37ecef1284d9701d6fb4e39e)
then