Thread: Cantor ternary set

Are there any irrational algebraic numbers in the Cantor ternary set?

Originally Posted by bobbyk

Are there any irrational algebraic numbers in the Cantor ternary set?

Interesting question. I don't know the answer (yet - I hope), but as web
searches have turned up nothing relevant its either trival or the proof of
(non-)existence of such is obscure.

My question is: In what context has this question been asked? Is this
homework, research or something else?

RonL

Tried a calculation, and saw that it can contain no n-th roots. This is too harsh a condition, so I bet it doesn't contain any algebraic irrationals at all. Don't have a complete proof, though.

Originally Posted by CaptainBlack

Interesting question. I don't know the answer (yet - I hope), but as web
searches have turned up nothing relevant its either trival or the proof of
(non-)existence of such is obscure.

My question is: In what context has this question been asked? Is this
homework, research or something else?

RonL

No, it's not homework or research. It's just something I've wondered about
for many years and have asked a number of mathematicians with no answers.

bobbyk

Originally Posted by bobbyk

Are there any irrational algebraic numbers in the Cantor ternary set?

The Wikipedia article on Normal Number* has something to say about
this:

"David H. Bailey and Richard E. Crandall conjectured in 2001 that every
irrational algebraic number is normal; while no counterexamples are known,
not a single irrational algebraic number has ever been proven normal in any
base"

Now if the Bailey-Crandall conjecture is true then Cantors ternary set would
contain no irrational algebraic numbers, as in base-3 the ternary set contains
no base-3 normal numbers.

(Note the Wikipedia article refers to normality but the Bailey-Crandall paper
refers to absolute normality which means normality in all bases)

RonL

* A number is normal to base-b if the base-b expansion contains every
sequence of digits of length with relative frequency . Which
essentialy means all digits, and strings of digits appear in the expansion of
with the frequency that they would be expected in a random string.

Thanks for noticing this conjecture! I hadn't heard of it. So my question is
still unanswered, but is probably in the negative, since no counterexamples
have been found. I was going to ask Ulam about this, but he died before I
could get around to it

What is a ternary set?
I was not able to find it on Wikipedia however it did have a Cantor set.

Originally Posted by ThePerfectHacker

What is a ternary set?
I was not able to find it on Wikipedia however it did have a Cantor set.

Its the same thing

RonL

Originally Posted by bobbyk

Thanks for noticing this conjecture! I hadn't heard of it. So my question is
still unanswered, but is probably in the negative, since no counterexamples
have been found. I was going to ask Ulam about this, but he died before I
could get around to it

I'm impressed by the name dropping, how did you come to know Ulam?

RonL

Sorry about the name dropping! I didn't know Ulam. I'm not even a mathematician. He was just one of my heroes.