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Conjecture:

For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$.

Questions:

  1. Has this conjecture been proved, refuted or neither?

  2. If proved:

    Is there an estimate of the minimum length of $q_b$ containing all the digits?

    For example, I would expect something like $2b$ or $b^2$ for any given $q_b$.

  3. If not refuted:

    I suppose that it is not true for transcendental numbers. Is that correct?

    How can we construct a transcendental number $q_b$ which does not contain all the digits?

Thanks

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For the last point: yes, as you seem to know, it fails for transcendental numbers: for example, Liouville's constant is transcendental, but only has $1$'s and $0$'s in its decimal expansion. For more information, read about general Liouville numbers. –  Geoff Robinson Jul 6 at 9:00
    
@GeoffRobinson: Liouville's constant? Is that from the proof of the fact that some problems cannot be solved on a Turing machine? Isn't that number given in base $2$ (in which case, it does contain all the digits)? –  barak manos Jul 6 at 9:04
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Assuming you are talking about initial sequences of digits: for the second point, as for any $b$ and any $n$ there are irrational algebraic numbers whose representation in base $b$ starts with $n$ zeros, there is no such bound. –  Stefan Kohl Jul 6 at 9:07
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The Liouville's constant I refer to is $\sum_{n=1}^{\infty}10^{-n!}.$ –  Geoff Robinson Jul 6 at 9:55
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I just want to remark since no one has mentioned it yet: there is a much stronger conjecture that every irrational algebraic real number is normal in every base. –  Bill Mance Jul 6 at 19:46
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2 Answers 2

The conjecture has been neither refuted nor proved. The state of the art, as far as I know, is contained in the papers of Adamczewski and Bugeaud, in which they show that anything with a very low complexity decimal expansion cannot be an algebraic irrational. The complexity is the function $c_x(n)$ giving the number of blocks of length $n$ in the decimal expansion of $x$ (or any base). They show that if there exists a $k$ such that $c_x(n)\le kn$ for all $n$, then $x$ is either rational or transcendental. Of course, it's conjectured that $c_x(n)=10^n$ for all algebraic irrationals $x$. Your condition would be implied by the conjecture $c_x(n)>9^n$ for all algebraic irrationals $x$.

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You might also look at questions 114905, 114758, 99039, where very similar questions are asked. –  Anthony Quas Jul 6 at 13:12
    
Thanks. Did you mean "cannot be irrational algebraic"? –  barak manos Jul 6 at 13:14
    
right - fixed now. –  Anthony Quas Jul 6 at 13:17
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What about the number in base 10 that has decimal expansion using only digits 1 and 2 in the following pattern: 0.121221222122221 ... That is the digit 1 occurs always alone: 2's occur in blocks of increasing length. This has no periodicity and is not a rational number (and misses many digits of the base 10 system)

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For this to be relevant to the conjecture, the number you constructed would have to be algebraic, but it is transcendental because it is $2/9 - \sum_{n=1}^\infty (1/\sqrt{10})^{n(n+1)}$. The latter term is related to a theta function value and was proved to be transcendental. projecteuclid.org/… –  Douglas Zare Jul 6 at 9:49
    
I did not know this was transcendental. Then this answers 3rd part of the question by OP. Thanks, Douglas Zare for such a detailed information and pointing out the reference. But I am still not able to figure out how to see that given number is the sum of the series you have described, also not able to see the connection to the Theta function paper you have cited. I'll think about it. –  P Vanchinathan Jul 6 at 10:16
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Thanks. From the comments to my question, I realized that both section #2 and section #3 were pretty easy to answer (kind of dumb questions to begin with I suppose). @Geoff Robinson gave a good example answering section #3, similar to yours I think. –  barak manos Jul 6 at 10:42
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Has this been done for $\sqrt D$ in arbitrary base? –  Lev Borisov Jul 6 at 12:20
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