Bounds for rational approximations to square roots
A consequence of Dirichlet’s approximation theorem is that for any real number , there exist infinitely many rational approximations that satisfy
In particular, the convergents of the continued fraction for satisfy this inequality. Hurwitz’s theorem gives a slightly tighter bound and states that there exist infinitely many rational approximations that satisfy
Surprisingly, this bound is tight, since for the golden ratio since for any there exist only finitely many solutions to .
It turns out that in general, there are bounds on how well rational numbers can approximate algebraic numbers. This is the basis of Liouville’s theorem, which was used to construct the first known transcedental numbers.
In Frits Beukers’ “Getaltheorie voor Beginners”, he briefly mentions an explicit bound for : any rational approximation to satisfies
He proves this by supposing that and deriving a contradiction. Multiplying both sides with gives
where the last inequality follows from
Finally, multiplying by gives
Now and are integers, so the left side is integer. The only integer with absolute value less than one is zero. However, this implies that so , which is a contradiction since is irrational.
This bound can obviously be improved, since there are places where inequalities are used that are not tight. For example, we use , while is much closer to than to .
We can derive a tighter bound by supposing for some and following the same steps until. We end up with
Now, we want to pick so that the right hand side becomes one. So we solve and find the positive solution . By this, we have found the tighter bound .
In fact, we can generalize this for any irrational :
Theorem: Let be a natural number that is not a square. Then any satisfies
Proof: Set and suppose there exists a with
Multiply both sides by to get
If we substitute in and simplify we see . So the right side of this equation equals . Multiplying both sides by gives us
Like before, this is a contradiction: it implies that , so , so , so . However, by assumption is not a square, so is irrational. So this is a contradiction.
I wonder if this can be generalized to th roots and numbers of the form .
From a high-level perspective we have used the expression
to measure how well a rational number approximates a real number. We can now rephrase some results in terms of :
- For any there are infinitely many with .
- For any natural number that is not a square root we have