On the support of a non-autocorrelated function on a hyperbolic surface

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 $e^{\frac{r}{2}}(r+1)^{-1}$.