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What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)?

What is the probability that a randomly chosen number from the set of computable numbers is period(number)?

For definition of period(number),please see www.maths.ed.ac.uk/~aar/papers/kontzagi.pdf‎.

Further more,if we replace period number with exponential periods,what are the probability in the question above?

The set of computable numbers includes set of periods or exponential periods ,what is the reverse?Is it true?or almost true in probability sense?

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    $\begingroup$ Since these are countable sets of reals, you should say a little more what you mean by "randomly chosen". $\endgroup$
    – Joel David Hamkins
    Mar 30, 2014 at 12:14
  • $\begingroup$ @JoelDavidHamkins,thank you Joel,I will think over it to make a clarification $\endgroup$
    – XL _At_Here_There
    Mar 30, 2014 at 13:13

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This paper by Yoshinaga shows that periods are contained within a proper subclass of the computable reals. Another proof is given in this paper by Tent and Ziegler. (Disclaimer: I haven't read either paper myself.) Whether the results in these papers answer your questions or not depends, as Joel points out, on what you mean by "randomly chosen".

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  • $\begingroup$ Thanks a lot, Denis.Yoshinaga has partially answered my question by an approach I has just begun to try,that is the computational complexity of a number.I just have read the abstract. $\endgroup$
    – XL _At_Here_There
    Mar 30, 2014 at 13:08
  • $\begingroup$ Tent and Ziegler just say in abstract that they have proved theorems presented by Yoshinaga.And a greeting,how are you doing recently? $\endgroup$
    – XL _At_Here_There
    Mar 30, 2014 at 13:11
  • $\begingroup$ You're welcome, and greetings to you as well. $\endgroup$
    – Denis Hirschfeldt
    Mar 30, 2014 at 16:20
  • $\begingroup$ @ DenisHirschfeld,I am wondering if there has been or possibly will be any result about constructing irrational algebraic numbers with degree exceeding 2 by constructive mathematics.I intend to think that the coefficients continued fraction expansions of irrational algebraic numbers with degree exceeding 2 are all unbounded.I had just read Stuart's paper on a related generalization of this question.He(Prof.Kurtz)has proven that the related generalization of this question is undecidable,but he original question is still open. $\endgroup$
    – XL _At_Here_There
    Aug 23, 2014 at 10:54
  • $\begingroup$ The period numbers are in the class of computable irrational numbers with lower computational complexity.It is very interesting to know that most transcendental numbers in number theory are in this class,which is in the lower level in the hierarchy of computable numbers $\endgroup$
    – XL _At_Here_There
    Aug 23, 2014 at 11:01

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