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Approximation by fractions
Posted:
Jul 5, 2014 4:21 PM


How to approximate a real by a fraction? "In one direction", the answers are wellknown. 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



