Suppose X/Γ is an arithmetic locally symmetric space of noncompact type (with the natural metric induced by the Killing form of the isometry group of X), and let ξ be a point on the visual boundary of X. T.Hattori showed that if each horoball based at ξ intersects every Γ-orbit in X, then ξ is not on the boundary of any Q-split flat in X. We prove the converse. (This was conjectured by W.H.Rehn in some special cases.) Furthermore, we prove an analogous result when Γ is a nonarithmetic lattice.