  Let $f$ be a non-negative square-integrable function on a finite volume
hyperbolic surface $\Gamma\backslash\mathbb{H}$, and assume that $f$ is
non-autocorrelated, that is, perpendicular to its image under the operator of
averaging over the circle of a fixed radius $r$. We show that in this case the
support of $f$ is small, namely, it satisfies $\mu(supp{f}) \leq
(r+1)e^{-\frac{r}{2}} \mu(\Gamma\backslash\mathbb{H})$.
  As a corollary, we prove a lower bound for the measurable chromatic number of
the graph, whose vertices are the points of $\Gamma\backslash\mathbb{H}$, and
two points are connected by an edge if there is a geodesic of length $r$
between them. We show that for any finite covolume $\Gamma$ the measurable
chromatic number is at least $e^{\frac{r}{2}}(r+1)^{-1}$.
