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Let $\mu (r)>2$ be the irrationality measure of a transcendental number $r$, and consider the following set of points $P \in\mathbb{R}$:

$P=\{r\in \mathbb{R}: \mu(r)=Constant\}$

Is this set a fractal, and If so, then what is it's dimension?

Matt Calhoun
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  • The known upper bound of e.g. $\mu (\zeta (3))$ is $5.513891$, but it might be improved. So what does it means $\mu (r)=$ Constant $=5.513891$ in $P$? – Américo Tavares Nov 17 '10 at 23:15
  • I would say that we should distinguish between two cases: $\mu(r)=\mu (\zeta (3))$ and $\mu(r)=5.513891$. I'm slightly confused by this comment, because if $\mu (\zeta (3))$ is an irrational number, then it's impossible to write it in decimal notation in any case. – Matt Calhoun Nov 18 '10 at 15:00
  • Correction: ... what does it mean ... . $\zeta(3)$ is indeed an irrational number with an irrational measure less or equal to $5.513891...$. And if $\zeta(3)$ is not a transcendental number? (Though I believe one day someone will prove it is). – Américo Tavares Nov 18 '10 at 15:25
  • If $\zeta (3)$ is algebraic, then $\mu(\zeta(3))=2$. – Matt Calhoun Nov 18 '10 at 18:12
  • Yes, I know. But what does the condition $\mu(r)=$ Constant mean, when applied to $r=\zeta(3)$? What is the value of the Constant that would make $r=\zeta(3)$ be in $P$? – Américo Tavares Nov 18 '10 at 20:31
  • Measure of irrationality from The Springer Online Encyclopaedia of Mathematics http://eom.springer.de/M/m063260.htm – Américo Tavares Nov 18 '10 at 21:07
  • thanks for the link. I am still trying to understand precisely what issue/problem you are identifying, would you mind explaining this in more detail please. Am I correct that you are uncertain that there are any real numbers for which $\mu{r}$ is strictly greater than 2, but less than infinity, otherwise I dont see the significance of $\zeta(3)$? – Matt Calhoun Nov 19 '10 at 20:34
  • Is your question a theoretical one? 2) It is difficult for me to understand the property $\mu(r)=$ Constant, except that the finite Constant is greater than or equal to 2. 3) The example I gave has no particular significance. 4) When only an upper bound greater than $5.513891$ was known (it was $12.417820\dots$, according to Apéry's proof), would you say that $\mu(\zeta(3))=5.513891\dots$? Of course, not, although now you can say it is.
    • – Américo Tavares Nov 19 '10 at 21:48
  • I think I finally understand your point, which is that $\mu(\zeta(3))$ is not known, and therefore should not be used as the constant $C$ in $\mu(r)=C$ because that is ambiguous. I agree of course, and I didn't see this as an issue because I was just working with arbitrary constant real numbers, which $\mu(\zeta(3))$ certainly is, even if it's unknown. – Matt Calhoun Nov 19 '10 at 22:54
  • Perhaps to answer (1) I should try and motivate my question a bit more: i just learned about irrationality measure last week, and I think it's really cool. It's called a "measure", and the most basic geometrical shape is the circle, so after thinking about these $P$ sets, and not finding any references online, it seemed like $P$ must have a fractal structure. Also, as soon as I see power law equations with non-integer exponents, the first thing I think of is fractals. It seemed like there might be a nice proof that this set is fractal and maybe even a nice way to compute its dimension. – Matt Calhoun Nov 19 '10 at 22:56