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.]