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A while back I wrote this question on PPCG.SE about the numbers I termed Pirrational numbers.

They are defined as follows:

Let $P_i$ be the $i$th Pirrational number for some $i \in \mathbb{N}_0$ with $P_0 = \pi$.

To generate $P_{i+1}$:

  1. List the digits of $P_i$ as a sequence of positive integers, ignoring duplicates.
    • e.g. $3, 1, 4, 15, 9, 2, 6, 5, 35,...$ (OEIS A064809) for $P_0$.
  2. Let $s_j$ be the $j$th term in this sequence for $j \in \mathbb{N}$.
  3. Create $P_{i+1}$ as the number with digits $d_1.d_2d_3d_4d_5...$ where $d_j = (s_j)\text{th digit of }P_i$.
    • Thus, for example, $P_1 = 4.3195...$ since $4$ is the third digit of $P_0$, $3$ is the first digit, etc.

The answers to the code challenge show that it's very difficult to get accurate values of even the first few Pirrational numbers. No one could calculate the first digit of $P_4$.

There are many approximations of $\pi$, can the Pirrational Numbers be approximated in similar ways? Are there any methods of approximation that would be useful in this case or are things too "digit-based" so to speak?

(I am mainly concerned with base 10 here, but of course Pirrationals exist in other bases.)

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I don't think you can approximate it. You are performing a permutation mapping on the digits, where the indices are generated by the previous element of the sequence. So, you are mapping sequences of digits - approximations only make sense if the "importance" of numbers diminishes, but here, an early digit can already call for an accurate digit from far away in the sequence. This way, a change very late in the previous number can affect the very front of the next one.

I have no idea how to prove it, but this mapping could be formally represented as a chaotic mapping, meaning that a small change results in a completely different number. Essentially you have a random number generator that utilizes mixing to produce essentially an uncorrelated sequence of digits.

However, asymptotic growth of the sequences $(s_j)$ in general could help you quantify how quickly you many digits of the previous term you need to get one digit of the next one.

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