{"paper":{"title":"Projections of planar sets in well-separated directions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Tuomas Orponen","submitted_at":"2015-04-27T18:05:32Z","abstract_excerpt":"First, let $K \\subset B(0,1) \\subset \\mathbb{R}^{2}$ be a set with $\\mathcal{H}_{\\infty}^{1}(K) \\sim 1$, and write $\\pi_{e}(K)$ for the orthogonal projection of $K$ into the line spanned by $e \\in S^{1}$. For $1/2 \\leq s < 1$, write $$E_{s} := \\{e : N(\\pi_{e}(K),\\delta) \\leq \\delta^{-s}\\}, $$ where $N(A,r)$ is the $r$-covering number of the set $A$. It is well-known -- and essentially due to R. Kaufman -- that $N(E_{s},\\delta) \\lessapprox \\delta^{-s}$. Using the polynomial method, I prove that $$ N(E_{s},r) \\lessapprox \\min\\left\\{\\delta^{-s}\\left(\\frac{\\delta}{r}\\right)^{1/2},r^{-1}\\right\\}, \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07189","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}