# Invariance of the (Liouville-Roth) irrationality measure under rational Möbius transformations

For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the (possibly empty) set of all real exponents $s > 0$ such that the inequality $$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s}$$ holds true for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$ (with the convention that $\sup(\emptyset) := \infty$).

Now, fix $\tilde{x} \in \mathbb{R}$ and let $M: \mathbb{R} \cup \{\infty\} \to \mathbb{R} \cup \{\infty\}$ be the Möbius transformation $$x \mapsto \left\{\begin{array}{ll} (ax+b)/(cx+d) & \text{if }x \ne \infty \\ a/c & \text{if }c \ne 0 \text{ and }x = \infty \\ \infty & \text{if }(c = 0 \text{ and }x = \infty) \text{ or }(c \ne 0 \text{ and }x = - d/c)\\ \end{array}\right.,$$ where $a,b,c,d \in \mathbb Q$ and $ad-bc \ne 0$. It is then possible to prove that $\mu(\tilde{x}) = \mu(M(\tilde{x}))$ provided that $c\tilde{x}+d \ne 0$, and here come my questions:

Q1. Is there a "standard" reference for this? I checked Mahler's Lectures on Diophantine Approximations and Schmidt's Diophantine Approximations and Diophantine Equations, but the result doesn't seem to be mentioned in either of them. Q2. From the historical point of view, who was the first one who proved the result? Maybe J. Liouville, in his early work in diophantine approximation theory?

Thank you beforehand for any help.

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See section 2.8 in Heights in diophantine geometry by Bombieri and Gubler, particularly Prop. 2.8.19. –  Vesselin Dimitrov Sep 26 '13 at 22:55
Better: see the original source, K. K. Choi and J. D. Vaaler, Diophantine approximation in projective space: cecm.sfu.ca/~choi/paper/metric.pdf . –  Vesselin Dimitrov Sep 26 '13 at 23:07
Thank you, Vesselin, all the more that Choi and Vaaler's paper points to Cassels' An introduction to diophantine approximation (Ch. I, Sect. 3, p. 11) for the basic case of ${\rm PGL}(2,\mathbb{Q})$ and approximations to reals by rational numbers. –  Salvo Tringali Sep 26 '13 at 23:48
I looked closer to Choi and Vaaler's results and Cassels' corollary on p. 11 of his book, and this morning, on a second thought, I don't see the relation with my questions. Those results do not apply to the Liouville-Roth irrationality exponent, but to the alternative measure of irrationality that we get by replacing $n^s$ with $s^{-1}n^2$ in the OP (which is relevant, in particular, in the study of badly approximable numbers). What do I miss? –  Salvo Tringali Sep 27 '13 at 6:18
Take $N=M=2$ and $\mathbb{x} :=(1:x)$, $\mathbb{y} := (1:m/n)$ in Thm. 3, with the projective distance defined in (1.2). Your question follows since $|\mathbb{x} \wedge \mathbb{y}| = |x-m/n|$. So you may quote the paper of Choi and Vaaler as a reference to a much more general setting. –  Vesselin Dimitrov Sep 27 '13 at 9:41