Estimating the fractal dimension of sets determined by nonergodic parameters

Given fixed and irrational \begin{document} $0 , consider the billiard table \begin{document} $B_{α}$ \end{document} formed by a \begin{document} $\frac{1}{2}×1$ \end{document} rectangle with a horizontal barrier of length \begin{document} $α$ \end{document} emanating from the midpoint of a vertical side and a billiard flow with trajectory angle \begin{document} $θ$ \end{document} . In 1969, Veech introduced two subsets \begin{document} $K_{0}≤ft(θ)$ \end{document} and \begin{document} $K_{1}≤ft(θ)$ \end{document} of \begin{document} $\mathbb{R}/\mathbb{Z}$ \end{document} that are defined in terms of the continued fraction representation of \begin{document} $θ∈\mathbb{R}/\mathbb{Z}$ \end{document} , and Veech showed that these sets have Hausdorff dimension \begin{document} $0$ \end{document} when \begin{document} $θ$ \end{document} is rational. Moreover, the set \begin{document} $K_{1}≤ft(θ)$ \end{document} describes the set of all \begin{document} $α$ \end{document} such that the billiard flow on \begin{document} $B_{α}$ \end{document} in direction \begin{document} $θ$ \end{document} is nonergodic. We show that the Hausdorff dimension of the sets \begin{document} $K_{0}≤ft(θ)$ \end{document} and \begin{document} $K_{1}≤ft(θ)$ \end{document} can attain any value in \begin{document} $≤ft[0, 1]$ \end{document} by considering the continued fraction expansion of \begin{document} $θ$ \end{document} . This result resolves an analogue of work completed by Cheung, Hubert, and Pascal in which they consider, for fixed \begin{document} $α$ \end{document} , the set of \begin{document} $θ$ \end{document} such that the flow on \begin{document} $B_{α}$ \end{document} in direction \begin{document} $θ$ \end{document} is nonergodic.