{
  "annotation_id": "9f73f1b7-8043-4766-8171-ad94962d90cb",
  "date_created": "2026-02-17T05:53:19.849000Z",
  "date_modified": "2026-02-17T05:53:19.849000Z",
  "file_hash": "f3a325c5df1c2700e9094d0cea98bade2b4e60b7518d7e4794aabe8631025272",
  "private": false,
  "record": {
    "abstract": "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.",
    "arxiv_id": "2601.09202",
    "authors": [
      "Longhui Li"
    ],
    "categories": [
      "math.CA",
      "math.DG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "On unions of geodesics and projections of invariant sets",
    "url": "https://arxiv.org/abs/2601.09202",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "0f210eb7-8cea-4eb0-910c-4d700744b158",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}