The digits of $\pi$ uniform distribution non-standard analysis, and its partial limit

I was wondering whether there has been any investigation into the distribution of the decimals of $\pi$ using non-standard analysis. For instance, can it proven that the frequency distribution is uniform in the strictest possible sense (where by this I mean that since non-standard analysis treats infinities as actual entities like finite quantities, the actual frequency and not just the relative frequency of each digit is precisely the same)? If so, I was wondering if this might explain its partial limit, if it has one (if absolute arithmetic relative frequency convergence requires place selection rules and a violation of randomness – there have been some papers recently arguing about whether tests of randomness appear to be violated given large numbers of the digits). This is based on one my previous conjectures concerning convergence in the finite, place selection rules and absolute infinite convergence in the arithmetic sense.

By a partial limit I am asking whether the sequence is such that there is some $n$, such that for all relative frequencies calculated over the first $K$ digits, where $K>n$, the distribution is within some error bound of being uniform, for all $K$, and where as $K$ increases this error bound (although not necessarily the error/deviation in each individual sequence) gets shorter. Whilst Fursternburg's theorem might suggest that there will always (or almost always) be infinite sequences of the same digit (and of any digit or pattern) one cares to name, do these appear? And if so, will they only occur after a significantly larger infinite number string of uniformly distributed digits beforehand so that the effect of such large strings of the same digit gets washed out, in the limiting relative frequency (so that the partial limit is not violated – or really infinite limit by this stage is perturbed/disturbed)?

Essentially, what I am asking is whether it would, say, be Abraham Wald-von Mises random immunity to gambling or place selections – can I determine a place selection so that for $n$th (or perhaps more complicated function) I can pick out an infinite sequence which is comprised of all and only one digit or, more simply, of some relative frequency different from $\pi$. Or rather can one find some function for any $i$th digit, that will give you what its value is simply as a function of all and only the digits beforehand (forgetting that actual analytic computation)?

Moreover, is it proven that $\pi$ is uniformly distributed – and if so, in the arithmetic non-standard sense or in the real analysis sense where it may diverge by some infinitesimal proportion from ten percent, by which we say it converges in any case?

The digits of $\pi$ uniform distribution nonstandard analysis, and its partial limit - ResearchGate. Available from: https://www.researchgate.net/post/the_digits_of_pi_uniform_distribution_nonstandard_analysis_and_its_partial_limit [accessed Jun 9, 2016].

  • $\begingroup$ It is widely accepted that $\pi$ is normal, but the truth is that we do not even know whether all digits appear infinite many times. It is still possible that only $2$ digits appear infinite many times. But the known digits of $\pi$ are very well equally-distributed. It is believed that this is the case for all digits. It is hard to imgaine that the normality of $\pi$, $e$ or any irrational algebraic number will ever be actually solved for a single base. $\endgroup$ – Peter Jun 9 '16 at 11:32

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