Order of error for rational approximation of irrationals

[Liedragged]

Hi, I have to approximate an irrational number x by rationals r = p/q.

Let ε>0 in ℝ, then, for almost all x exist α and r in (x-ε,x+ε) such that q ≈ c(x) ε^-α, c(x) in ℝ?

I know, from Hurwitz theorem (and a conseguence) that α>2, if exists.

 

[morphism]

I'm confused. What exactly are you asking? You have a question mark there, but no real question. One interpretation of what you could be asking is answered immediately by Hurwitz's theorem.

 

[Liedragged]

I'll try to be more precise...

Let x be an irrational, let ε>0,

Let q = inf_A q'

Where A = {r rational | r=p/q', |x-r|<ε}.

For small ε, I want to know if there exist an estimate (not an inequality) for almost all x of q(ε)

I know, from Hurwitz theorem that:

q(ε)<1/sqrt{5} ε^{-2}

I know, also that:

q(ε) ≈ ε^{-2} (order minus 2)

only for a countable set (measure = 0).

It is possible that exist α in ℝ such that:

q(ε) =c(x) ε^{-α} + o( ε^{-α}) for almost all x, c(x)>0 ?

 

[dodo]

Widening the question a little... is there any reason why continued fractions cannot be used? They'd give the best rational approximation for the smallest denominator, and the error bounds are well defined.

 

[Liedragged]

Yes, there is a reason, I have to do only a theoretical analysis (not a real estimate of numerical error), so the word "error" have a restricted validity in this thread.

There are "different degrees" of irrational numbers? What is the distribution of the α's? Is it concentrate around a single value?

I need this for an argument on Hamiltonian systems, I cannot say more...