Re: Arbitrary strings of digits in the decimal display of irrational numbers
- From: Daniel Lichtblau <danl@xxxxxxxxxxx>
- Date: Mon, 18 Jun 2007 19:47:08 -0700
On Jun 18, 8:41 pm, Juergen Barsuhn <jw.bars...@xxxxxxxxxxx> wrote:
Let me return to this topic once more. In previous contributions
"Klueless" on 02.06.2007 ans
G.A.Edgar also on 02.06.2007
have essentially contributed to this question and helped me to overcome
some misconceptions.
Let me first try to formulate the present status, I hope I get it right
this time.
We consider the set A of irrational numbers in [0,1] in their decimal
representation. The measure of A is 1 and A is not countable. Be p the
property of a certain irrational number that you can find any arbitrary
finite string of digitd (0,1,2,...9) somewhere in the decimal display of
that number. All members of A that exhibit the property p build the
subset B of A. The measure of B is 1 and B is uncountable.
The set C = A minus B is of measure 0, "but" also not countable. For a
member of C it is possible to find (at least) one finite string of
digits that does not occur in the display of that member of C. (For
diifferent memebers of C these strings might be different.)
"Klueless" has given a nice construction of an uncountable subset of B
in his posting.
(As you may find a certain finite string s as substring of infinitely
many finite strings S1, S2, S3, ...., the property p means that you can
find any string s infinitely many times in the decimal display.)
It is unpleasing that of many irrational numbers like Pi - 3, exp(1) -
2, sqrt(2) - 1 it is so far not known, whether they belong to B or to C
- though at least for Pi - 3 there is a "strong belief" that it belongs
to B (I remember an "internet game" "Find your phone number in Pi !")
To me the question arises whether p is an "intrinsic" property of an
irrational number or is it an effect of the "interaction" of that
irrational number with the special representation with the basis 10
chosen here. To be more precise, let us talk in the section above of
the property p10 and the sets B10 and C10. We could look at the octal
representation of a member of B10 and ask whether it has the
corresponding property p8 of exhibiting all finite strings of octal
digits (0,1,2,3,...7). Or would it be possible that a member of A10
belongs to An, but to Bm - n and m being different bases of number
systems 2,3,4,....
If p were a property of an irrational number valid in any number system
(dual, octal, decimal, hexadecimal or whatsoever) one could hope to find
a property that is equivalent to p but in contrast to p could be checked
somehow in practise, in order to *prove* the presence or absence of p.
All the best Jurgen
This does not exactly handle the question you raise, but may give an
idea of where to look in the literature.
Your property p is closely related to a stronger property, b-
normality, where b is the base in question. A number 0<n<1 is b-
normal if, in it's base b representation, every finite string of
length k occurs with averge frequency (1/b)^k (average frequency being
defined via appropriate limit). This implies, among other things, that
all finite digit strings occur infinitely often.
It is known that for all b>=2 the set of b-normal numbers has measure
1. See:
http://mathworld.wolfram.com/NormalNumber.html
every base has measure 1, since the complement is a countable union ofFrom here it is a simple argument that the set of numbers normal to
sets of measure 0. For more information have a look at
http://mathworld.wolfram.com/AbsolutelyNormal.html
The notes at the URLs above give several good references for this
topic.
Daniel Lichtblau
Wolfram Research
.
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