Bounded-clustering conjecture for cocompact lattices

Establish the bounded-clustering property for every cocompact lattice in the relevant groups [?] as formulated by Sarnak, namely, prove that for each cocompact lattice [?] there exists a constant [?] such that the number of distinct traces in every unit interval (or, in the Kleinian case, every unit square) is uniformly bounded.

Background

The paper distinguishes the positive trace gap condition from Sarnaks bounded-clustering property. Positive trace gap implies bounded clustering, but the converse is not established by the methods developed in the paper: bounded clustering permits arbitrarily close pairs of traces, whereas positive trace gap rules them out.

Sarnaks bounded-clustering conjecture asserts that a lattice with uniformly bounded numbers of traces in translates of a unit interval is arithmetic; the paper notes that this conjecture has been proved for nonuniform Fuchsian lattices and for nonuniform lattices in the relevant two-dimensional and three-dimensional hyperbolic groups, but remains unresolved for cocompact lattices.