  Suppose X/Gamma 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 p be a point on the visual boundary of X. It was shown by T.Hattori
that if each horoball based at p intersects every Gamma-orbit in X, then p is
not on the boundary of any Q-split flat in X (where Q is the field of rational
numbers). We prove the converse. (This was conjectured by W.H.Rehn in some
special cases.) Furthermore, we prove an analogous result when Gamma is a
nonarithmetic lattice.
