To be clear, in the statement of the conjecture, we do not need f(q) to be a polynomial or “nice” function of q like 1/q or 1/q^2 — it may not even be an increasing function of q (so statements like “we want f(q) not to decrease much faster than 1/q^2” can be a bit misleading) — one can have arbitrary functions like, say, specifying that f(12)=1/4, f(20)=1/8, and for other q, f(q) = {0 if q is a multiple of 7 (so basically such denominators are never allowed), 1/3 if q does not contain the digit 9, 1/(q log q) otherwise}, etc. I imagine the fact that the proof has to work with arbitrary functions was part of the difficulty.

 Yes, f can be any function at all here, and indeed this allows for the sort of more complicated things you mention, and indeed this is where a lot of the difficulty lies. Duffin and Schaeffer, for whom the conjecture is named, had already proved it subject to a restriction on f that kinda-sorta requires it not to vary too wildly.(One other bit of precision that I glossed over: the conjecture is actually about approximations p/q for which p and q have no common factor and that genuinely makes a difference here.)

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