This is a whimsical question, motivated purely by curiosity rather than for any application.

We are all familiar with countless mathematical results which use Archimedes' constant $\pi$ either in their statement, or in their proof. However, in almost all cases that I am aware of, the precise numerical value of $\pi = 3.14159\dots$ does not play any significant role in the arguments; in some hypothetical alternate mathematical universe in which $\pi$ somehow managed to take on some different positive real numerical value, most proofs involving $\pi$ would continue to be locally valid despite the global inconsistency created.

So my question is whether there are any examples of mathematical theorems where a numerical bound on $\pi$, e.g., $\pi \geq 3$ or $\pi \leq 22/7$, or even much cruder bounds such as $\pi \geq 1$, actually played an important role in the proof, in that the proof would fail or require significant modification if such a bound was not known. (Here I exclude the really trivial bounds $0 < \pi < +\infty$, which certainly are implicitly used all over the place.) To put it another way: how many digits of $\pi$ do we need to know in order to do modern mathematics?

This can be contrasted with the analogous question for Euler's number $e=2.71828\dots$, where one can certainly think of places where it was important to know that, for instance, $e \geq 2$ (and certainly $e>1$ is **extremely** important!). But I can't think of many examples involving $\pi$; this constant seems to be a lot less "willing" to behave like a genuinely dimensionless quantity than say $e$, $\sqrt{2}$, or the golden ratio $\phi$, in that it rarely interacts in an additive fashion with such quantities, and its presence often cancels itself out at the end of a calculation.

To avoid some degenerate answers, let me exclude the following categories of theorems from this question:

**Theorems about $\pi$ itself**, such as Lindemann's theorem that $\pi$ is transcendental, or mathematical coincidences involving $\pi$ such as the Feynman point.**Theorems involving a lengthy numerical calculation involving expressions that contain $\pi$.**An example might be a PDE result which relied on some numerical estimation of an integral expression involving $\pi$, or an eigenvalue of an operator which also involved $\pi$ somehow. [I would waive this exclusion though if one could make a clear case that the numerical value of $\pi$ played a particularly decisive role in the calculation, as opposed to the appearance of $\pi$ merely being an artefact of one's normalization conventions (e.g., for the Fourier transform) that played no actual role in the final theorem.]**Theorems arbitrarily comparing numerical quantities, at least one of which involves $\pi$ implicitly or explicitly.**An example of this might be a result comparing the volume of a cube and a ball in some dimension, but without any motivation for why it would be interesting to know which one is larger.**Theorems in which a numerical bound involving $\pi$ was replaced with a slightly weaker numerical bound not involving $\pi$, purely to make the final conclusion look nicer.**(Suggested by JoshuaZ in the comments.)**Theorems that gratiutiously use $\pi$ in their proof, but for which the proof can be easily modified to avoid mention of $\pi$ (or related quantities).**(Suggested by Christian Remling in the comments; for instance, the statement that a square inscribes a circle of diameter equal to the sidelength can be proven directly without explicit mention of $\pi$.)

I'll start with one near-example to this question: in some sieve-theoretic applications in analytic number theory, it is useful to know that the average value of $|\sin x|$, namely $\frac{2}{\pi}$, is strictly less than one (see, e.g., the end of Section 4 of this blog post of mine on a Banach algebra proof of the prime number theorem). Ostensibly this is using the lower bound $\pi > 2$. However, one doesn't actually need to know about $\pi$ to see this fact, since it is immediate just from the inspection of the graph of $|\sin x|$ that its average value is going to be less than one! The fact that this average value also happens to be evaluated explicitly as $\frac{2}{\pi}$ is, as far as I know, inessential to the applications that I am aware of, and so the appearance of $\pi$ here is (almost literally) tangential. (Instead, one can view this argument as providing a proof of the lower bound $\pi>2$, and thus is basically a result of the first category of exclusions; one can also view this particular example as also belonging to the fifth category.)

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