Dichotomy for the Hausdorff dimension of the set of nonergodic directions

Given an irrational 0<λ<1, we consider billiards in the table P _λ formed by a \(\tfrac{1}{2}\times1\) rectangle with a horizontal barrier of length \(\frac{1-\lambda}{2}\) with one end touching at the midpoint of a vertical side. Let NE?(P _λ ) be the set of θ such that the flow on P _λ in direction θ is not ergodic. We show that the Hausdorff dimension of NE?(P _λ ) can only take on the values 0 and \(\tfrac{1}{2}\), depending on the summability of the series \(\sum_{k}\frac{\log\log q_{k+1}}{q_{k}}\) where {q _k } is the sequence of denominators of the continued fraction expansion of λ. More specifically, we prove that the Hausdorff dimension is \(\frac{1}{2}\) if this series converges, and 0 otherwise. This extends earlier results of Boshernitzan and Cheung.