{"paper":{"title":"Remarks on dimension of unions of curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Hyerim Ko, Sanghyuk Lee, Seheon Ham, Sewook Oh","submitted_at":"2022-08-05T16:43:56Z","abstract_excerpt":"We study an analogue of Marstrand's circle packing problem for curves in higher dimensions. We consider collections of curves which are generated by translation and dilation of a curve $\\gamma$ in $\\mathbb R^d$, i.e., $ x + t \\gamma$, $(x,t) \\in \\mathbb R^d \\times (0,\\infty)$. For a Borel set $F \\subset \\mathbb R^d\\times (0,\\infty)$, we show the unions of curves $\\bigcup_{(x,t) \\in F} ( x+t\\gamma )$ has Hausdorff dimension at least $\\alpha+1$ whenever $F$ has Hausdorff dimension bigger than $\\alpha$, $\\alpha\\in (0, d-1)$. We also obtain results for unions of curves generated by multi-parameter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.03272","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.03272/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}