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Topic: Approximation by fractions
Replies: 3   Last Post: Aug 5, 2014 8:09 AM

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Ilya Zakharevich

Posts: 152
Registered: 12/13/04
Approximation by fractions
Posted: Jul 5, 2014 4:21 PM
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How to approximate a real by a fraction?  "In one direction", the
answers are well-known.  However, it looks like if one asks ³the
opposite question², the popular knowledge is silent.

I suspect that the answer to this question is tramped to death by
specialists, but I have no clue *where* to look:

Conjecture.  There is a value of c (probably c approx 1) such that
given alpha, any
solution (p,q) to |p/q - alpha| < c/q^2 has either a form (mP,mQ), here
P/Q is
a continued fraction approximant to alpha obtained by cutting before
the
coefficient n with m^2 <~ n, or P' + tP, Q' + tP, here (P,Q) and
(P',Q') are sequential approximants, and |t| <~ 1.

 [Symbol <~ is less than in order of
 magnitude; see also ilyaz.org/fonts.]

Thanks in advance,
Ilya



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