Kontsevich and Zagier proposed a definition of a period (see for example http://en.wikipedia.org/wiki/Ring_of_periods ). The set of periods is countable, so not all of $\mathbb{C}$. I heard a rumour today that there is now a known explicit complex number which is not a period; does anyone know if this is true, or have more details?
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9$\begingroup$ I added an algebraic geometry tag. I quote from Kontsevich-Zagier: "It can be said without much overstretching that a large part of algebraic geometry is (in a hidden form) the study of integrals of rational functions of several variables." $\endgroup$– Kevin H. LinCommented Apr 6, 2010 at 0:05
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3 Answers
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Such a number is constructed in this article of M.Yoshinaga where it is proved that periods can be effectively approximated by elementary rational Cauchy sequences.
answered Apr 5, 2010 at 23:50
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2$\begingroup$ I haven't followed the details of that paper yet, but I would expect that it's simpler to prove that Chaitin's constant $\Omega$ is not a period, although computable numbers may be viewed as more explicit than $\Omega$. $\endgroup$ Commented Apr 6, 2010 at 0:32
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$\begingroup$ Tent and Ziegler also have a proof of Yoshinaga's theorem. arxiv.org/abs/0903.1384 $\endgroup$ Commented Apr 6, 2010 at 0:44
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$\begingroup$ @Douglas Zare: I'm guessing the article above is one of the first to give any sort of qualitative property of periods. And I think it makes sense that as an application he constructed a computable number that is "non-elementary", even though Chaitin's constant could have produced a faster counter example. This would probably be the difference of what M. Waldschmidt calls analog of Liouville vs. analog of Hermite in swc.math.arizona.edu/aws/08/slides/08WaldschmidtSlides5.pdf (slide no. 31) :) $\endgroup$ Commented Apr 6, 2010 at 1:04
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2$\begingroup$ Although it is clearly a right way to provide an example of non-period, by showing a certain nice approximativity of periods (and this was exactly Liouville's trick in 1844), I wonder whether the proofs are correct. I'll make a serious look only after the results appear in a serious journal. Sorry for my sceptisism. $\endgroup$ Commented Apr 6, 2010 at 8:39
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I haven't heard the latest about periods, but I want to point out a potential fallacy here. It's said very often that the easy proof of the existence of transcendental numbers (on cardinality grounds) is non-constructive. But, that's false! It is constructive. Given pen, paper and lots of time, I could extract from that argument the decimal expansion $0.a_1 a_2 \ldots$ of the transcendental number that the proof constructs.
See, for example, these comments of Joel David Hamkins.
I suspect that the same is true for periods: that there's an effective enumeration of them, so there's an algorithm for generating the decimal digits of a number that isn't a period.
answered Apr 5, 2010 at 23:02
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3$\begingroup$ Well, whether that's the right phrase or not, what I meant was that there exist an enumeration $p_1, p_2, ...$ of the periods and an algorithm that will churn out the 1st decimal digit of $p_1$, then the 1st of $p_2$, then the 2nd of $p_1$, then the 1st of $p_3$, then the 2nd of $p_2$, then the 3rd of $p_1$, etc. In that case, there is an algorithm that will churn out, for successive values of $n$, the $n$th decimal digit of $p_n$. So then a non-period is constructible. $\endgroup$ Commented Apr 6, 2010 at 0:08
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2$\begingroup$ OK. In that case, there is definitely such an enumeration and such an algorithm. The enumeration is obvious given your favorite enumeration of $\mathbb{Q}$, and the algorithm is your favorite numerical integration algorithm. $\endgroup$ Commented Apr 6, 2010 at 0:14
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2$\begingroup$ Good! So then, the importance of the article of Yoshinaga (cited by Gjergji) can't be that it simply constructs a non-period. $\endgroup$ Commented Apr 6, 2010 at 0:54
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2$\begingroup$ Boris, because it seems too easy. I mean, if I'm not mistaken then Kevin and I have just sketched an algorithm for constructing a real number that isn't a period, in a few minutes and using only the definition of period (no theory). Surely to an expert this would count as obvious. $\endgroup$ Commented Apr 6, 2010 at 14:10
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6$\begingroup$ Not that this matters much, but it's possible that Tom's focus on decimal digits introduces an annoying technicality. Namely, it's at least conceivable that some numbers in the list are so close to rational numbers that the decimal expansions contain enormously long finite strings of 9's, where the lengths of these strings grow faster than any computable function. Then your Turing machine will run out of gas trying to decide the values of the digits just before the strings of 9's. One can get around this by asking for sufficiently good rational approximations rather than decimal digits. $\endgroup$ Commented Sep 8, 2010 at 14:27
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I recently attended (and blogged about) a colloquium on precisely this topic in which your 'rumour' was mentioned. Apparently it has been conjectured that neither $e$ nor the Euler constant $\gamma$ are periods.
answered Apr 5, 2010 at 23:01
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1$\begingroup$ Kontsevich-Zagier already state these conjectures in their paper. They also state the conjecture that $1/\pi$ is not a period. $\endgroup$ Commented Apr 5, 2010 at 23:53
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$\begingroup$ Indeed, thanks -- I just checked. The speaker did not attribute the conjectures, but I think that a lot of what he said can be found in Kontsevich--Zagier. $\endgroup$ Commented Apr 6, 2010 at 0:55
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3$\begingroup$ And don't forget that a much simpler problem, the irrationality of $\gamma$, remains open. In this context showing that a given constant, like $e$ or $1/\pi$, or a Liuoville number, is a non-period looks like a big dream. $\endgroup$ Commented Apr 6, 2010 at 12:55
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$\begingroup$ Yes, I've heard of $\gamma$ spoken of as a "pain" by some well-known mathematicians. $\endgroup$ Commented Apr 6, 2010 at 13:31