Re: A weird question about pi

I don't remember where I heard this, but in a similar context, someone
remarked that there are only countably many "patterns" (since one would
presume a "pattern" can be described, and there are only countably many
descriptions), whereas there are uncountably many irrationals. If we
use this loose definition for "pattern", then almost all irrationals
have random distribution, and the set of those which do have a pattern,
is of measure zero.

Of course, a proper definition of "pattern" would need to be in place
to make this more rigorous.

One could extend the notion of "pattern" to allow uncountably many of
them, by considering distribution. For example, looking at all the
irrationals on (0,1) expressed as binary, let fn(x) be the ratio of 1's
to 0's at the nth binary digit, and consider f(x) = lim fn(x). One
might expect f(x) = 0.5 for most x, but certainly there exist
irrational x's with f(x) being all over the map. If say f(x) = 100, is
that not a "pattern" of some kind? If so, there are uncountably many
such "patterns". And of course there are irrationals where lim fn(x)
does not even exist.

.

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