Diophantic approximations

[dro...@gmail.com]

I have a question about approximations of irrational numbers
by rationals. Roughly speaking, if x is an irrational, then there
exist an infinitely many distinct fractions p/q such that
|x - p/q| < q^(-2). These fractions are given by the convergents
of the continued fraction expansion of x. The converse is also true:
If p/q is a fraction such that |x - p/q| < 0.5*q^(-2) than p/q
must be one of the convergents as above.

My question has to do with approximating x by fractions which, in some
sense,
have nothing to do with continued fractions. Is it possible? For
example, does there
exist an infinitely many fraction p/q such that

q^(-1.8) < |x - p/q| < q^(-1.5) ?

I am writing q^(-1.8) on the left side of the above inequality
to eliminate what's called the secondary convergents. These are
fractions
of the form (t*p_n + p_(n-1))/(t*q_n + q_(n-1)), where p_k/q_k
denotes the kth convergent of CF of x. These approximate x fairly
well, to within c*q(-2) for some c, I believe. I am interested in
something
*not* originating with the continued fraction apparatus. Thus, the
numbers
-1.8 and -1.5 above could be replaced by any other meaningful
quantities.

I will be grateful for any pointers.

Thanks in advance.

As ever,

Vlad

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* Vladimir Drobot
* Retired and gainfully unemployed
* http://www.vdrobot.com
* mailto:dro...@pacbell.net
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