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Let $f$ be a non-negative square-integrable function on a finite volumehyperbolic surface $\Gamma\backslash\mathbb{H}$, and assume that $f$ isnon-autocorrelated, that is, perpendicular to its image under the operator ofaveraging over the circle of a fixed radius $r$. We show that in this case thesupport 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 ofthe graph, whose vertices are the points of $\Gamma\backslash\mathbb{H}$, andtwo 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 measurablechromatic number is at least $e^{\frac{r}{2}}(r+1)^{-1}$.

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