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

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- Jan 28th 2006, 12:49 AMbobbykCantor ternary set
Are there any irrational algebraic numbers in the Cantor ternary set?

- Jan 28th 2006, 04:24 AMCaptainBlackQuote:

Originally Posted by**bobbyk**

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 - Jan 28th 2006, 07:52 AMRebesques
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.

- Jan 28th 2006, 01:27 PMbobbykQuote:

Originally Posted by**CaptainBlack**

for many years and have asked a number of mathematicians with no answers.

bobbyk - Jan 28th 2006, 02:14 PMCaptainBlackQuote:

Originally Posted by**bobbyk**

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 $\displaystyle N$ is normal to base-b if the base-b expansion contains every

sequence of digits of length $\displaystyle m$ with relative frequency $\displaystyle b^{-m}$. Which

essentialy means all digits, and strings of digits appear in the expansion of $\displaystyle N$

with the frequency that they would be expected in a random string. - Jan 28th 2006, 03:03 PMbobbyk
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 - Jan 28th 2006, 05:08 PMThePerfectHacker
What is a ternary set?

I was not able to find it on Wikipedia however it did have a Cantor set. - Jan 28th 2006, 10:31 PMCaptainBlackQuote:

Originally Posted by**ThePerfectHacker**

RonL - Jan 29th 2006, 03:03 AMCaptainBlackQuote:

Originally Posted by**bobbyk**

RonL - Jan 30th 2006, 01:36 PMbobbykUlam
Sorry about the name dropping! I didn't know Ulam. I'm not even a mathematician. He was just one of my heroes.