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Order of error for rational approximation of irrationals

  1. Mar 12, 2012 #1
    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.
     
  2. jcsd
  3. Mar 12, 2012 #2

    morphism

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    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.
     
  4. Mar 13, 2012 #3
    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 wanna 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 ?
     
  5. Mar 13, 2012 #4
    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.
     
  6. Mar 13, 2012 #5
    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...
     
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