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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\\}, \\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1504.07189","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-04-27T18:05:32Z","cross_cats_sorted":[],"title_canon_sha256":"de69c9da5dfe34553e4223645681b1cb3e31c05ab9629ee3a6e754466f57ed70","abstract_canon_sha256":"8dc56158d1812c671796db86ea617ae6cf3c864909293f077fed26c121e4d532"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:37.615333Z","signature_b64":"TIT//qNHBwoeoR11VUjJQ+Zsudo6ViZcifwcnwvzY3PKwSQq0o9b9CQeuoUcsUvFujKp2kqUr926soxeTd0YAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86081570df3668817ca4d1bbfbaafdf538b57789ee3d8095bcd717b63d63caad","last_reissued_at":"2026-05-18T01:16:37.614818Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:37.614818Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1504.07189","created_at":"2026-05-18T01:16:37.614906+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.07189v5","created_at":"2026-05-18T01:16:37.614906+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07189","created_at":"2026-05-18T01:16:37.614906+00:00"},{"alias_kind":"pith_short_12","alias_value":"QYEBK4G7GZUI","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"QYEBK4G7GZUIC7FE","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"QYEBK4G7","created_at":"2026-05-18T12:29:39.896362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U","json":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U.json","graph_json":"https://pith.science/api/pith-number/QYEBK4G7GZUIC7FE2G57XKX56U/graph.json","events_json":"https://pith.science/api/pith-number/QYEBK4G7GZUIC7FE2G57XKX56U/events.json","paper":"https://pith.science/paper/QYEBK4G7"},"agent_actions":{"view_html":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U","download_json":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U.json","view_paper":"https://pith.science/paper/QYEBK4G7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.07189&json=true","fetch_graph":"https://pith.science/api/pith-number/QYEBK4G7GZUIC7FE2G57XKX56U/graph.json","fetch_events":"https://pith.science/api/pith-number/QYEBK4G7GZUIC7FE2G57XKX56U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U/action/storage_attestation","attest_author":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U/action/author_attestation","sign_citation":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U/action/citation_signature","submit_replication":"https://pith.science/pith/QYEBK4G7GZUIC7FE2G57XKX56U/action/replication_record"}},"created_at":"2026-05-18T01:16:37.614906+00:00","updated_at":"2026-05-18T01:16:37.614906+00:00"}