Heaviness in Circle Rotations

We are concerned with describing the structure of the set of points in the unit interval which, when subjected to rotation by irrational alpha modulo one, for all finite portions of the orbit contain at least as many points in the bottom half of the interval as in the top half. Specifically, an inductive procedure for describing the set based on the continued fraction expansion of alpha is developed, leading into a discussion of the Hausdorff dimension of this set. Depending on the parameter alpha, all possible dimensions may be achieved, and the essential infimum (with respect to alpha) of this dimension is positive.