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

TL;DR

This paper demonstrates that non-autocorrelated functions on hyperbolic surfaces have small support and uses this to establish a lower bound on the measurable chromatic number of certain hyperbolic graphs.

Contribution

It introduces a bound on the support of non-autocorrelated functions on hyperbolic surfaces and applies this to derive chromatic number bounds for hyperbolic graphs.

Findings

Support of non-autocorrelated functions is exponentially small in radius r.

Lower bound for measurable chromatic number grows exponentially with r.

Provides a link between harmonic analysis and graph coloring on hyperbolic surfaces.

Abstract

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