Newsgroups: sci.math
Subject: Some BIG and SMALL sets of integers.
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: 11 Sep 92 17:18:04 +1200
Distribution: world
Organization: Department of Mathematics, University of Canterbury

The thread on the "size" of various sets of positive integers, reminded me of an
amazing result, which I read in "A Number for Your Thoughts", by S.P.Richards.
---

Call a subset of N, "BIG", if the sum of reciprocals of its elements is
infinite, and "SMALL", if finite. So any set with positive "density", as
defined in recent posts, is BIG; as can be proved without too much trouble.

But not vice versa. The set of primes is zero-density, but is still BIG;
whereas the set of powers-of-2 is zero-density and SMALL.

[ Extraneous query: are sets with no limiting density necessarily "big" ? ]
---

Anyway, Richards now looks at the decimal representation (groan) of integers,
and asks, how many have the "correct" number of "1"s.    i.e. how many
integers have *exactly* one tenth of their digits being a "1".
As it were, "1-average" numbers.    6733310002 is an example.

At first sight it doesn't seem like many, as for a start the integer would
have to have a multiple-of-10 number of digits. But there are quite a few
of these, in fact, and with a bit of work one can prove that the set of
integers with exactly 1/10 of their digits being "1" is BIG.

Of course there is nothing special about "1"; any other named digit would
easily give the same result. Now suppose we look at the proportions of two
named digits, say "1" and "2" (again, exactly which two is irrelevant); and
ask similarly, how many integers have exactly 1/10 of their digits being "1"
*and*  1/10 of their digits being "2".  What we might call "1-2-average" numbers.

Again, this time with quite a bit more work, which I will leave as an excercise
for the reader,  :)  one can show that the set of "1-2-average" numbers is
a BIG set.

So we find that there are "a lot" of 1-digit-average and 2-digit-average numbers.

Obviously, we ask the same about "3-digit-average" numbers, and so on all
the way up to 9 (equivalently, 10); and it seems obvious that they should all
be BIG sets of numbers as well.

BUT NO !!!!!!!
~~~~~~
Amazingly, as soon as we go to "3-digit-average" numbers, these are now
only a SMALL set of numbers.  (Again the proof left to the reader.)
That is, the set of numbers that have exactly 1/10 of their digits being
a "1"  *and*  1/10  being a  "2"  *and*  1/10  being a "3", is a SMALL set !
And as every 4-digit-average set is a subset of some 3-digit-average set,
these (and higher ones) are all SMALL sets as well.

IMHO, this is a truly startling result. One could believe that there might
be a difference between 1-digit-average and the rest, maybe; or between
9-digit-average and those below, perhaps. But that the distinction should come
between  *2*  and  *3* ;  that is really weird !
                                  ~~~~~~~~~~~~
---
As a final incidental remark: changing the base from 10 to any other base
from 2 upwards, doesn't essentially change things. This is especially intriguing
for base 2 and base 3, as then we get that the set of numbers with the "correct"
proportion of *all* digits is a BIG set !

-------
A coda.  Perhaps those familiar with the recurrence/transcience of
random walks on lattices in 2/3 dimensions might not find this *quite* so
startling; and if you search for the proofs of the above results, you may
find that they hold, "for the same reason" as do the random walk results.

-------------------------------------------------------------------------------
           Bill Taylor              wft@math.canterbury.ac.nz
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 Free will- the result of chaotic amplification of quantum events in the brain.
Galaxies- the result of chaotic amplification of quantum events in the big bang.
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