{"paper":{"title":"A discretised projection theorem in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Tuomas Orponen","submitted_at":"2014-07-24T12:16:43Z","abstract_excerpt":"The main result of this paper is that for any $1/2 \\leq s < 2 - \\sqrt{2} \\approx 0.5858$, there is a number $\\sigma = \\sigma(s) < s$ with the following property. Let $\\delta > 0$ be small, assume that $A \\subset [0,1]$ is a $(\\delta,1/2)$-set, and that $E \\subset [0,1]$ contains $\\gtrsim \\delta^{-\\sigma}$ roughly $\\delta^{s}$-separated points. Then there exists a number $t \\in E$ such that $A + tA$ contains $\\gtrsim \\delta^{-s}$ $\\delta$-separated points.\n  For $\\sigma = s$, this is essentially a consequence of Kaufman's well-known bound for exceptional sets of projections. Our proof consists "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6543","kind":"arxiv","version":2},"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"}