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Given $n \in \mathbb{N}$, let $p(n) \in \mathbb{N}$ be such that $n/p(n)$ is the best approximation of $\pi$ (denoted as $\tilde{\pi}_n$). I have two main questions:

  1. Is the sequence $\{p(n)\}$ a known sequence? If so, is there a closed-form expression for it?
  2. Can we establish an upper bound for the absolute error of each approximation, $|\pi - \tilde{\pi}_n|$?

Some observations: From numerical computations for $1 \leq n \leq 2000$, we observe the following:

enter image description here

The results suggest that $\max |\pi - \tilde{\pi}_n|$ decays faster than exponentially. However, I am curious if there is a rigorous way to prove this behaviour. I also like the elegant "arches", so I wonder what can be said about those.

Out of curiosity, this is the error for Euler's number $e$, for $1\leq n\leq 10000$ (the "pillars" seem to have a higher length variance).

enter image description here

In fact, these pillars of irrationality will always occur with some periodicity for any irrational number, never reaching zero. Is this also a definition of an irrational number?

sam wolfe
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    Google "continued fraction". There's a lot of good stuff for you to discover out there. – JonathanZ Nov 27 '24 at 14:26
  • You can search in the box for approximation \pi (with dollar signs around the \pi) and find many questions on the subject. – Ross Millikan Nov 27 '24 at 14:34
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    Do your arches have their "pillars" at $n=355$, $n=710$, $n=1065$, $n=1420$, $n=1775$ by any chance? The curves between those are interesting. – Chris Lewis Nov 27 '24 at 14:37
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    This question has been closed because of a duplicate but I would argue that this question touches on different areas than the duplicate. Specifically, the duplicate is about best rational approximations to $\pi$ with denominator is a certain range. This question is about the best approximation for any numerator $n$, and how the errors in these change with $n$. The pattern in these errors alluded to in this question is not covered at all in the duplicate. – Chris Lewis Nov 27 '24 at 14:44
  • @ChrisLewis They do! Related? However these sequences are different after $n=6035$. – sam wolfe Nov 27 '24 at 17:08
  • This and this might also be related. – sam wolfe Nov 27 '24 at 17:30
  • Fun fact: $p(n)$ is not just the nearest integer to $\frac{n}{\pi}$. It seems to be the case most of the time, but $n=14$ is a counterexample; the next counterexamples are at $n=344$ and $n=51819$. OEIS doesn't seem to have this sequence. If you look at the fractional part of $\frac{n}{\pi}$ for these $n$ they are very close to $0.5$ (which makes sense). – Chris Lewis Nov 27 '24 at 17:47

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