On unions of geodesics and projections of invariant sets

Let $M$ be a $d$-dimensional complete Riemannian manifold and let $\pi: SM \to M$ denote the canonical projection from the unit tangent bundle. We prove that if $E \subset SM$ is a set that invariant under the geodesic flow with Hausdorff dimension $\dim_{\mathcal{H}} E \ge 2(k-1)+1 +\beta$ for some integer $1 \le k \le d-1$ and some $\beta \in [0,1]$, then the projection $\pi(E)$ satisfies $\dim_{\mathcal{H}} \pi(E) \ge k + \beta$. In other words, this yields a lower bound on the Hausdorff dimension of unions of geodesics in $M$. Our theorem extends a result of J. Zahl concerning unions of lines in $\mathbb{R}^d$. The proof relies on the transversal property of geodesics, an appropriate $(k+1)$-linear curved Kakeya estimate, and the Bourgain-Guth argument.