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    $\begingroup$ I was going to answer Apéry's proof of irrationality of $\zeta(3)$. But I looked at it and in fact it does not fit here. Some asymptotics in there involve $\pi$, but the ones needed for the irrationality proof involve only $e$ and $\sqrt2$. (The $\pi$s all cancel.) $\endgroup$
    – Gerald Edgar
    Commented May 12 at 16:58
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    $\begingroup$ Does the Basel problem count? I have read, that Euler knew the numerical value of $\pi^2/6$ so that he could guess and prove it: eulerarchive.maa.org/hedi/HEDI-2003-12.pdf $\endgroup$
    – mathoverflowUser
    Commented May 12 at 16:58
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    $\begingroup$ Somewhat along the lines of your near-example, Bourgain's Sum-free subset theorem goes through a painful case analysis where the inner product of various trigonometric series is taken against test functions and one needs a few digits of pi to conclude that the results are sufficient to conclude the argument. See 3.15 through 3.23 at link.springer.com/article/10.1007/BF02774027. $\endgroup$
    – Mark Lewko
    Commented May 12 at 16:58
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    $\begingroup$ @MarkLewko Ah, yes, this is exactly the kind of thing I had in mind! It might be fun to see exactly what bounds on $\pi$ are needed here in order to obtain Bourgain's final conclusion (which I already had a fondness for, as it is a good candidate for minimizing the ratio (degree of improvement over trivial bound)/(mathematical strength of author)). $\endgroup$
    – Terry Tao
    Commented May 12 at 17:25
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    $\begingroup$ Searching for uses of the bounds in Lean, I can see that $\pi>3$ is used for proving that the ring of integers of $\mathbb{Q}[\zeta_3]$ and $\mathbb{Q}[\zeta_5]$ is a PID. Not sure if it fits the requirements. The upper bound I found only in here where it seems to be used for upper bounds on diagonal Ramsey numbers. We could search for such uses in other proof assistants. $\endgroup$
    – Daniel Weber
    Commented May 12 at 17:31