ioannis

In 1937 Mahler proved that Champernowne's number was transcendental. In fact he
proved the much stronger result that if p(x) is a polynomial with integer
coefficients then the real number obtained by concatenating the integers p(1),
p(2), p(3), ... to get

0.p(1)p(2)p(3)...

is transcendental. Champernowne's number is the special case where p(x) = x.

Does anyone know if there's any reason why the stronger result with a(n) being a
(non-trivial) sequence was not proved?

For example, if a(n)=floor[exp(n)], I would expect the number:

0.a(1)a(2)a(3)...

to also be transcendental. I can see it, but I cannot prove it.

A polynomial p(x) with integer coefficients is just a special sequence.

Any references or pointers towards that direction are appreciated.

Many thanks,
--
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/

g. a. edgar

Can you also "see" this one: floor(10^n/3) ??

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

ioannis

I believe I said "non-trivial" sequences (which in this context is taken to mean
sequences with terms with no repeating patterns).

Otherwise feel free to speculate about a(n)=floor[exp(n)].
--
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/

[ Moderator's note:
"Non-trivial" needs to be replaced by a precise definition
in order to get something that might be provable. "Not having
a pattern that I can see" will not work. As for the original
question, the obvious answer is that if Mahler had known how
to prove a stronger result he would have done so.
-RI ]

ol3

Maybe the base of the exponential function must be other than any power
of the base used for representation?

The moral of the story: You need to look at proofs for specific kinds
of functions in order to avoid pitfalls.

--OL

bo198214

I (and probably you too) can not see a repeating pattern in the
sequence of decimal figures of sqrt(2), despite it is not
transcendental.

ioannis

Probably true, but my question is concerned with a more general construction.

For example, if a(n)=floor(sqrt(2)*10^(n-1)), then it appears to me as though
0.a(1)a(2)a(3)... is transcendental.
--
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/

[ Moderator's note:

The point is that in most cases you can't tell whether a number
is transcendental by looking at its digits. It's almost
certainly true that the number Ioannis mentions here is
transcendental, but there's no proof. Conjectures of this
sort are very easy to make, but very hard to prove. Unless
somebody has something really new to say, I'm closing this
thread.

-RI]

ioannis

Before Robert closes the thread, let me make my conjecture as best as I can:
Call a "morphism" a pattern of digits a(n) which is in one to one correspondence
with the naturals and further it has the property a(n)+a(m)=a(n+m).

For example, the decimal patterns of C_2 (Champernowne's base 2 pattern) are a
morphism, because they can be mapped directly onto the Naturals and they satisfy
a(n)+a(m)=a(n+m). So are the patterns of C_b, where b is any base in {2,..10}.
Simply associate the number 1 with its base b pattern, 2 with its base b
pattern, and so on.

Although a general non-trivial sequence gives a one to one correspondence
between N and its terms {a(n)}, n\in N, the pattern it generates
(0.a(1)a(2)a(3)...) is not necessarilly a morphism. Take a(n)=floor(exp(n)), for
example: a(1)+a(2)=2+7 =/= a(3)=20.

The case I have and I am interested in is the sequence a(n) described in this
webpage:

http://ioannis.virtualcomposer2000.com/math/Naturals.html

Although it looks that this a(n) is not a morphism
(a(1)+a(2)=102+101022=/=101021010222=a(3)), on another level it looks like *it
is* a morphism, since the word for a(n) is essentially nothing but a string
representation of all the preceding naturals, 012...n-1, therefore a(n) is
basically a different representation of n itself.

On some level then (which sorry, I cannot put to words) c(n) is essentially
Champernowne's expansion, expressed using different symbols. Therefore it looks
as if A is transcendental iff C_{10} is.

I am sorry, this is the best I can do to describe the problem. I seem to be
lacking the description tools in this case.
--
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/