Rational approximation
Contents
Introduction
The main theme of this article is the question how well a given real number can be approximated by rational numbers. Of course, since the rationals are dense on the real line, we, surely, can make the difference between
and its rational approximation
as small as we wish. The problem is that, as we try to make
closer and closer to
, we may have to use larger and larger
and
. So, the reasonable question to ask here is how well can we approximate
by rationals with not too large denominators.
Trivial theorem
Every real number can be approximated by a rational number
with a given denominator
with an error not exceeding
.
Proof
Note that the closed interval has length
and, therefore, contains at least one integer. Choosing
to be that integer, we immediately get the result.
So, the interesting question is whether we can get a smaller error of approximation than . Surprisingly enough, it is possible, if not for all
, then, at least for some of them.
Dirichlet's theorem
Let be any integer. Then there exists a rational number
such that
and
.
Proof of Dirichlet's theorem
Consider the fractional parts . They all belong to the half-open interval
. Represent the interval
as the union of
subintervals
. Since we have
fractional parts and only
subintervals, the pigeonhole principle implies that there are two integers
such that
and
belong to the same interval. But then the difference
differs by less than
from some integer number
:
. Dividing by
, we get
.
Corollary
If is irrational, then there are infinitely many irreducible fractions
such that
.
Proof of the corollary
For each , find a (possibly reducible) fraction
with
such that
. Let
be the same fraction as
but reduced to its lowest terms. It is clear that
, so it remains to show that among the fractions
there are infinitely many different ones. But the distance from the
-th fraction to
does not exceed
, which can be made arbitrarily small if we take large enough
. On the other hand, if the fractions
were finitely many, this distance couldn't be made less than the distance from the irrational number
to some finite set of rational numbers, i.e., less than some positive constant.
Discussion
The Dirichlet's theorem can be generalized in various ways. The first way is that one can try to approximate several numbers simultaneously by fractions with common denominator. The exact statement is as follows.
If and
is an integer, then there exists an integer
with
and integers
such that
The proof is essentially the same except instead of considering numbers
,
one has to consider
vectors
,
in the unit cube
divided into
equal subcubes.
Another remark that can be useful in some problems is that, if is irrational, then you can find infinitely many solutions of the inequality
with the denominator
contained in any given arithmetic progression
(
) if the constant
(depending on the progression) is large enough. To prove it, first, find infinitely many irreducible fractions
satisfying
. Then, for each such fraction, find two integers
such that
and
. Now note that
and
are relatively prime, so we can find some integer
such that
. Replacing
and
by their remainders
and
modulo
, we get a positive integer
satisfying
and
. Thus, setting
, we get
.
Liouville Approximation Theorem
We can generalize Dirichlet's theorem as follows: If is an algebraic number of degree
, then there are only finitely many rational numbers
satisfying the following inequality:
. This gives us the following corollary:
is a transcendental number, known as Liouville's constant.