  We solve Boshernitzan's problem of characterization (in terms of so called
Furstenberg systems) of bounded sequences that are orthogonal to all uniquely
ergodic systems. Some variations of Boshernitzan's problem involving
characteristic classes are considered. As an application, we characterize
sequences orthogonal to all uniquely ergodic systems whose (unique) invariant
measure yields a discrete spectrum automorphism as those satisfying an averaged
Chowla property.
