Question about normality and number bases
Joona I Palaste
Just a simple question. If a number's representation in one number base
is normal (i.e. it contains all finite digit strings in that number
base), is it automatically also normal in all other number bases?
My intuition tells me the answer to this is "yes", but I do not know how
to prove it. Thanks for your answers!
--
/-- Joona Palaste (pal...@cc.helsinki.fi) ---------------------------\
| Kingpriest of "The Flying Lemon Tree" G++ FR FW+ M- #108 D+ ADA N+++|
| http://www.helsinki.fi/~palaste W++ B OP+ |
\----------------------------------------- Finland rules! ------------/
"Bad things only happen to scoundrels."
- Moominmamma
G. A. Edgar
<pal...@cc.helsinki.fi> wrote:
> Hello,
> Just a simple question. If a number's representation in one number base
> is normal (i.e. it contains all finite digit strings in that number
> base), is it automatically also normal in all other number bases?
> My intuition tells me the answer to this is "yes", but I do not know how
> to prove it. Thanks for your answers!
The answer is: No, not necessarily.
One can construct numbers normal in base 2 but not base 3 and such.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
G. A. Edgar
> > Hello,
> > Just a simple question. If a number's representation in one number base
> > is normal (i.e. it contains all finite digit strings in that number
> > base), is it automatically also normal in all other number bases?
> > My intuition tells me the answer to this is "yes", but I do not know how
> > to prove it. Thanks for your answers!
>
> The answer is: No, not necessarily.
> One can construct numbers normal in base 2 but not base 3 and such.
OK, here is a reference:
J.W.S. Cassels, "On a problem of Steinhaus about normal numbers",
Colloquium Mathematicum 7 (1959) 95--101
There are numbers x in whose base 3 expansion the digit 2
never occurs but are normal in every base b which is
not a power of 3. (In fact, this is true for "almost all"
x in whose base 3 expansion the digit 2 never occurs,
"almost all" taken in an appropriate measure.)
David C. Ullrich
wrote:
>Hello,
>Just a simple question. If a number's representation in one number base
>is normal (i.e. it contains all finite digit strings in that number
>base),
The definition of "normal" is that it contain all finite digit
strings, _with_ the right _frequency_ (ie in base 10, 1/10 of the
digits are 0, 1/10 of the digits are 1, 1/100 of the two-digit strings
are 00, etc.) (Where the frequency is the limit as N -> infinity of
the frequency among the first N digits.)
>is it automatically also normal in all other number bases?
No. It's easy to give a counterexample if we're talking about
a weaker version of normality, where we just require that
each _digit_ occur with the right frequency. For example
in base 2, 1/3 = 0.01010101..., which contains half 0's and
half 1's, so 1/3 is weakly normal in base 2, but obviously
not in base 3.
It's not so easy to give a counterexample with the actual
definition of normal as above, involving all finite strings
of digits. But it "must" be that there exist examples.
For example: An element of the middle-thirds Cantor set
is a number between 0 and 1 that has a base-3 expansion
using only 0's and 2's, no 1's; so no element of the
Cantor set can be normal base 3. But if you choose the
0's and 2's for an element of the Cantor set at random
I would be very surprised if you did not almost surely
get a number that's normal in base 10 (and I would
not be too surprised to learn that a random "contruction"
like this was the easiest way to show a counterexample
exists.)
>My intuition tells me the answer to this is "yes", but I do not know how
>to prove it. Thanks for your answers!
>
>--
>/-- Joona Palaste (pal...@cc.helsinki.fi) ---------------------------\
>| Kingpriest of "The Flying Lemon Tree" G++ FR FW+ M- #108 D+ ADA N+++|
>| http://www.helsinki.fi/~palaste W++ B OP+ |
>\----------------------------------------- Finland rules! ------------/
>"Bad things only happen to scoundrels."
> - Moominmamma
David C. Ullrich
David C. Ullrich
So the Ullrich Conjecture was proved 43 years before I conjectured
it (more or less as I suspected). I think this makes it a remarkable
conjecture - any bozo can make conjectures about things that are
actually not known...
>--
>G. A. Edgar http://www.math.ohio-state.edu/~edgar/
David C. Ullrich
Gerry Myerson
ull...@math.okstate.edu wrote:
=> On Mon, 01 Jul 2002 09:03:44 -0400, "G. A. Edgar"
=> <ed...@math.ohio-state.edu> wrote:
=>
=> >>
=> >> > Hello, Just a simple question. If a number's representation in
=> >> > one number base is normal (i.e. it contains all finite digit
=> >> > strings in that number base), is it automatically also normal in
=> >> > all other number bases? My intuition tells me the answer to this
=> >> > is "yes", but I do not know how to prove it. Thanks for your
=> >> > answers!
=> >>
=> >> The answer is: No, not necessarily. One can construct numbers
=> >> normal in base 2 but not base 3 and such.
=> >
=> >OK, here is a reference: J.W.S. Cassels, "On a problem of Steinhaus
=> >about normal numbers", Colloquium Mathematicum 7 (1959) 95--101
=> >
=> >There are numbers x in whose base 3 expansion the digit 2 never
=> >occurs but are normal in every base b which is not a power of 3.
=> >(In fact, this is true for "almost all" x in whose base 3 expansion
=> >the digit 2 never occurs, "almost all" taken in an appropriate
=> >measure.)
=>
=> So the Ullrich Conjecture was proved 43 years before I conjectured it
=> (more or less as I suspected). I think this makes it a remarkable
=> conjecture - any bozo can make conjectures about things that are
=> actually not known...
Considerably more is known. Given any two multiplicatively independent
integers (that is, no power of the one is a power of the other) there's
a continuum of reals normal to the one base and not normal to the other.
I think that's a result of Wolfgang Schmidt.
I think it goes even farther - given any two (finite?) sets of
integers, no product of powers of elements of the one equal to
a product of powers of elements of the other, etc., etc.
--
Gerry Myerson (ge...@mpce.mq.edi.ai) (i -> u for email)
Nico Benschop
>
> >
> > > Hello,
> > > Just a simple question. If a number's representation in one
> > > number base is normal (i.e. it contains all finite digit strings
> > > in that number base), is it automatically also normal in all
> > > other number bases? My intuition tells me the answer to this is
> > > "yes", but I do not know how to prove it.
> > > Thanks for your answers!
> >
> The answer is: No, not necessarily.
> One can construct numbers normal in base 2 but not base 3 and such.
> OK, here is a reference:
> J.W.S. Cassels, "On a problem of Steinhaus about normal numbers",
> Colloquium Mathematicum 7 (1959) 95--101
>
> There are numbers x in whose base 3 expansion the digit 2
> never occurs
Re: P.Erdo"s, R.Graham (conjecture, 1980):
2^n base 3 (natural n) contains no digit 2 iff n=2 or n=8.
http://americanscientist.org/Issues/Comsci01/Compsci2001-11.html
(page 5: "Turning to Ternary Dust") -- NB
David C. Ullrich
Makes the UC even more remarkable... heh-heh.
>Given any two multiplicatively independent
>integers (that is, no power of the one is a power of the other) there's
>a continuum of reals normal to the one base and not normal to the other.
>I think that's a result of Wolfgang Schmidt.
Huh. Of course if you asked a person to guess whether this was so he
probably would have guessed yes. Anyway, one thing that seems
interesting about the result of Cassels cited is that almost
every element of the Cantor set is normal in base whatever. Given
two multiplicatively independent b_1 and b_2, is there a "natural"
measure on the set of all numbers not normal in base b_1 that
makes almost all of them normal in base b_2?
>I think it goes even farther - given any two (finite?) sets of
>integers, no product of powers of elements of the one equal to
>a product of powers of elements of the other, etc., etc.
>--
>Gerry Myerson (ge...@mpce.mq.edi.ai) (i -> u for email)
David C. Ullrich
G. A. Edgar
<ge...@mpce.mq.edi.ai.i2u4email> wrote:
Some searching SciMath shows that it is here...
Schmidt, Wolfgang M.
Über die Normalität von Zahlen zu verschiedenen Basen.
Acta Arith. 7 1961/1962 299--309.
Divide the integers > 1 into two classes, such that for every n,
all powers of n are in the same class as n. Then there exist
(a continuoum of) numbers x, normal for all bases in the first class,
non-normal for all bases in the second class.
Gerry Myerson
ull...@math.okstate.edu wrote:
=> ...one thing that seems interesting about the result of Cassels cited
=> is that almost every element of the Cantor set is normal in base
=> whatever. Given two multiplicatively independent b_1 and b_2, is
=> there a "natural" measure on the set of all numbers not normal in
=> base b_1 that makes almost all of them normal in base b_2?
Well, you're getting beyond where I have any confidence in my knowledge
on the subject, but I do recall that some of the work on these things
has involved constructing natural-but-unusual measures on these sets.
The authors involved have included Andy Pollington, Bill Moran,
and/or Gavin Brown, and the measures are called Riesz product measures.