I was wondering whether there has been any investigation into the distribution of the decimals of pi using nonstandard analysis. For instance can it proven that the frequency distribution is uniform in the strictest possible sense (whereby 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 requres 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 fursternburgs theorem might suggest 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 that such large strings of the same digit, gets washed out, in the limiiting relative frequency (so that the partial limit is not violated- or really infinite limit by this stage is perturbed/disturbed)?
the sequence is infinite...
https://www.researchgate.net/publication/303802949_Finding_Pi?ev=prf_pub
Article Finding Pi
What I am essentially 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 nth (or perhaps more complicated function) i can picked 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 ith 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
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