hi,
recently Joth Tupper posted a message with the following
"I seem to recall a non-intuititive theorem about rational approximations to
numbers (this is from c. 1968). If you can approximate a number too closely,
then it is transcendental. S.Lang wrote a book on trancendental numbers and
degrees around 1973 and a precise statement might be there.
Does anyone recall this?"
Yes!
This is the result that Liouville proved to show that Liouville's number
must be transcendental. The result is known, funnily enough as Liouville's
theorem.
It says that if \alpha is an algebraic number of degree n \geq 2, then there
exists a constant c(\alpha)>0 such that
| \alpha - p/q | > c(\alpha)/|q|^{n}
always holds.
Or if you prefer, for any m>n,
| \alpha - p/q | < |q|^{-m} ---- call this (*)
has only finitely many solutions.
At the root of the proof is the extremely "profound" fact that there are no
integers between 0 and 1 (or rather that the integers are discrete):
Let f(x)=x^n+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} be the "minimal polynomial" for
\alpha over the rationals. Let p/q be a rational number.
Then
q^{n}f(p/q) is an integer, it cannot be zero because f(x) is the minimal
polynomial for \alpha, therefore
|q^{n}f(p/q)| \geq 1.
f(x) =(x-\alpha)g(x) where g(x) is some other polynomial, so
|q^{n}(p/q-\alpha)g(p/q)| \geq 1 or
|p/q-\alpha| \geq |1/(g(p/q)q^{n})|.
One can estimate g(p/q) quite easily and use this estimate to get the value
for c(\alpha) mentioned above.
You prove that the Liouville number (or any similar type number) is
transcendental from this result is by noting that the rational
approximations are getting more and more accurate at each step (you are
basically adding lots more zeros each time). But Liouville's theorem forbids
this, if the number is algebraic. Thus the number is transcendental.
It can be formulated somewhat different yet again (in terms of "heights" of
numbers) and it is an extremely important result that is used almost
everywhere in diophantine analysis and transcendental number theory (so
ultimately most proofs in these areas depend upon cleverly showing that some
"deep" result depends on the fact that there are no integers between 0 and 1
:-)).
Liouville proved his theorem around 1840. It has since been improved.
Thue (1909) showed that you can weaken the condition on m in (*) above to
m>n/2
Siegel (1922) shows that m>2\sqrt{n} suffices
Dyson and Gelfond independently showed in the 40's that m>\sqrt{2n} suffices
and finally (sort of) Roth in 1954 showed that m>2 suffices.
Roth won a Fields medal for this (and some other) work.
Does that answer your questions?
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