From: harshec@mail.auburn.edu (Eric C. Harshbarger)
Subject: My other Christmas Question
Reply-To: harshec@mail.auburn.edu
Organization: Auburn University
Date: Thu, 21 Jan 1993 21:19:13 GMT


Thanks to all who responded to my 'HARSH' question about primes (q.v.
'Question about Primes'). Since that was well received, I thought I`d
post the other question I came up with during the holidays-- it's a
little harder to describe, so bear with me. I hope it proves interesting.

Consider the interval I=[0,1].
It is fact that every element of I may be represented by an infinite
binary (Base 2) expansion; examples:

.0101010101.... = 1/3
.0111111111.... = 1/2 or
.1000000000.... = 1/2
etc...

DEFINE set S(2\0) to be all points in [0,1] which may be represented by expansions in Base 2 which exclude 0 after some finite point in the expansion (i.e. after some point in the expansion only 1s appear). Note that this `finite point` is not fixed; it can vary depending on the expansion-- as long as after *some* point 0s no longer appear.

DEFINE S(2\1) as the set of points in [0,1] whose Base 2 expansions exclude 1 after some finite point.

DEFINE set S2 as [S(2\0) union S(2\1)].

Now, consider representations of elements of [0,1] in Base 3 (tertiary expansions). Again, with no restrictions, all numbers in [0,1] are represented.

DEFINE S(3\0) as the set of all points in [0,1] which can be represented by Base 3 expansions which exclude 0 after some finite point.

Similarly, DEFINE S(3\1) & S(3\2) as the sets of points in [0,1] which can be represented in Base 3 which exclude 1 & 2 respectively after some finite point.

DEFINE S3 as [S(3\0) union S(3\1) union S(3\2)]. Thus, S3 consists of all points in [0,1] whose Base 3 expansions exclude at least one of the numbers 0,1,or 2 after some finite point. In other words, the only points _not_ in S3 are ones whose tertiary expansions require endless use of all three digits.

Note that the point 1/3 is in S3, but not S2.

Continuing...

Consider Base 4 and in a similar fashion define sets S(4\0), S(4\1), S(4\2), and S(4\3)-- for example S(4\2) is the set of all points of [0,1] which can be represented by a Base 4 expansion which exclude the number 2 after some finite point in the expansion.

Also, DEFINE S4 as [S(4\0) union S(4\1) union S(4\2) union S(4\3)].

Now, continue this process for _every_ positive integer base to obtain sets S5, S6, S7, S8, S9, S10, S11, and so forth.

Notice that each set S# has measure 0 (it is the union of a finite number of sets which are each measure 0 by arguments similar to why the familiar Cantor 1/3 set is of measure 0).

Finally, having obtained S# for every positive integer (a countable collection), DEFINE S to be the union of all S#, # a +integer.

O.K., set S must also have measure 0 since it is the union of a countable collection of sets each of measure 0.

***So my question is... `What`s not in S`? I guess I really mean, can anyone give me a description of a number in some 'fairly nice' way and guarantee that it does not lies in S (in other words, for every Base the representative expansion in that Base continuely requires the use of all numbers in that Base-- no number stops showing up; or, another way, you can pick any finite point in the expansion and every number of the Base will show up after that point). 

Since S is of measure 0, and [0,1] has measure 1, there have to be plenty of points not in S but in [0,1].

Also, I said a 'fairly nice` description... I think I can get one in terms of monotone decreasing compact intervals of [0,1], but I'm not happy with that. I'm more interested in numbers described in terms of expansions or whatnot.

For example... is PI in S or not? Can anyone prove that the statement 'For every Positive Interger Base PI requires all numbers of that Base to continually show up in its Base expansion` is True or False?

I hope I've explained the question clearly (and correctly). As before, if the answer is terribly obvious or the question trivial, I apologize.

			Thanks,
				E.C. Harshbarger.