Let (X,\mu,T) be an ergodic probability measure preserving system on a metric space X, and let U be a non-empty open subset of X. Consider the (\mu-null) set of points in X whose trajectory completely misses U. Is it true that this exceptional set has Hausdorff dimension less than the dimension of X? And does the same hold for the set of points that visit U less frequently than prescribed by Birkhoff's Ergodic Theorem? The affirmative answer to the first question has been conjectured for actions on homogeneous spaces and proved in several special cases, for example when X is compact or has rank one. I will sketch a proof of a fairly general, although not optimal, answer – for arbitrary \textrm{Ad}-diagonalizable flows on irreducible quotients of semisimple Lie groups. Two main ingredients are effective mixing and the method of integral inequalities for height functions on X. Joint work with Shahriar Mirzadeh.