# average value of j-invariant at infinity

Let $$\xi\in\mathbb{R}$$ and consider the average value (with respect to hyperbolic length) of the $$j$$-invariant ($$j(z)=q^{-1}+744+196884q+\ldots$$, $$q=e^{2\pi iz}$$) along a geodesic aimed at $$\xi$$: $$\lim_{T\to\infty}\frac{1}{T}\int_0^Tj(\xi+e^{-t}i)dt.$$ This limit, when it exists, defines an $$SL_2(\mathbb{Z})$$-invariant function.

The only obvious (at least to me) values of $$\xi$$ for which the above limit exists are real quadratic numbers, where the path of integration is asymptotic to the geodesic (closed on the modular surface) joining $$\xi$$ with its Galois conjugate, i.e. the integrand is asymptotically periodic. For some numerical investigations in this case, see M. Kaneko http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/43observations_j_values.pdf.

[One could also use some rational function of $$j$$ to measure how far from some set of points a geodesic trajectory is, a variation on http://www.math.ucla.edu/~wdduke/preprints/modularbilliards.pdf.]

However, it seems that the limit above could exist for some other badly approximable numbers (those irrational real numbers $$\xi=[a_0;a_1,a_2,\ldots]$$ whose continued fraction partial quotients $$a_n$$ are bounded, equivalently those $$\xi$$ for which the path $$\{\xi+e^{-t}i : t\geq0\}$$ is bounded on the modular surface).

I wondered whether this average had been investigated elsewhere, if there were reasons why it wouldn't exist for other badly approximable numbers, and whether or not it is interesting/tractable as a measure of approximation. At first glance it doesn't seem to carry any more information than the continued fraction expansion, but maybe there's some modular magic I don't understand floating around.