Order of error for rational approximation of irrationals

In summary, the conversation discusses approximating an irrational number x by rationals and the existence of an estimate for q(ε). It mentions Hurwitz's theorem and the possibility of an estimate involving α. The speaker also mentions the use of continued fractions and the distribution of α's, but explains that the topic is related to Hamiltonian systems and cannot be discussed further.
  • #1
Liedragged
6
0
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.
 
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  • #2
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.
 
  • #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 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 ?
 
  • #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.
 
  • #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...
 

Related to Order of error for rational approximation of irrationals

What is the order of error for rational approximation of irrationals?

The order of error for rational approximation of irrationals is a measure of how close the rational approximation is to the actual value of the irrational number. It is represented by the difference between the approximation and the actual value, and is typically expressed as a fraction or decimal.

How is the order of error calculated?

The order of error is calculated by taking the absolute value of the difference between the rational approximation and the actual value of the irrational number and dividing it by the actual value. This value is then multiplied by 100 to convert it to a percentage.

What is the significance of the order of error in rational approximation?

The order of error is significant because it gives an indication of the accuracy of the rational approximation compared to the actual value of the irrational number. A lower order of error indicates a closer approximation, while a higher order of error indicates a larger discrepancy between the approximation and the actual value.

Can the order of error be reduced?

Yes, the order of error can be reduced by using a higher degree of approximation, such as using a larger numerator and denominator for the rational approximation. Additionally, using more terms in the continued fraction expansion of the irrational number can also help reduce the order of error. However, it is not possible to completely eliminate the order of error in rational approximation of irrationals.

What is the relationship between the order of error and the degree of approximation?

The order of error and the degree of approximation have an inverse relationship. This means that as the degree of approximation increases, the order of error decreases. However, increasing the degree of approximation also means using larger numbers and more terms, which may not always be practical or efficient.

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