{"paper":{"title":"Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"If dim E >1, dim E + dim F >2 and F has equal Hausdorff and packing dimension, then some pinned distance set Delta_y(E) has positive Lebesgue measure.","cross_cats":["math.AP","math.CO"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2026-03-16T14:20:56Z","abstract_excerpt":"Suppose $E, F$ are Borel sets in the plane, $\\dim_{\\mathcal{H}} E>1$, $\\dim_{\\mathcal{H}} E+\\dim_{\\mathcal{H}} F>2$, and $F$ has equal Hausdorff and packing dimension. We prove that there exists $y\\in F$ such that the pinned distance set $$\\Delta_y(E):=\\{|x-y|:x\\in E\\}$$ has positive Lebesgue measure. In particular, it settles the regular case of the distance set problem in the plane. The main ingredients of the proof consist of a multi-scale Good-Bad decomposition and a multi-scale Mizohata-Takeuchi-type estimate with arbitrary small power-loss."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"There exists y∈F such that Δ_y(E) has positive Lebesgue measure, given dim_H E>1, dim_H E + dim_H F>2, and F has equal Hausdorff and packing dimension.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that F has equal Hausdorff and packing dimensions, which enables the multi-scale Good-Bad decomposition and Mizohata-Takeuchi estimates to control the distance set without extra losses.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Under dim_H E >1, dim_H E + dim_H F >2 and F regular (equal Hausdorff and packing dimensions), there exists y in F such that the pinned distance set Δ_y(E) has positive Lebesgue measure.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If dim E >1, dim E + dim F >2 and F has equal Hausdorff and packing dimension, then some pinned distance set Delta_y(E) has positive Lebesgue measure.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a12a8a1c1ddc5271fca40fb5366cf02fcae0a74037608e6692ff4fd115ae1d68"},"source":{"id":"2603.15328","kind":"arxiv","version":3},"verdict":{"id":"4973fefb-539f-413b-9a61-5cf109904fcc","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T10:25:21.289653Z","strongest_claim":"There exists y∈F such that Δ_y(E) has positive Lebesgue measure, given dim_H E>1, dim_H E + dim_H F>2, and F has equal Hausdorff and packing dimension.","one_line_summary":"Under dim_H E >1, dim_H E + dim_H F >2 and F regular (equal Hausdorff and packing dimensions), there exists y in F such that the pinned distance set Δ_y(E) has positive Lebesgue measure.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that F has equal Hausdorff and packing dimensions, which enables the multi-scale Good-Bad decomposition and Mizohata-Takeuchi estimates to control the distance set without extra losses.","pith_extraction_headline":"If dim E >1, dim E + dim F >2 and F has equal Hausdorff and packing dimension, then some pinned distance set Delta_y(E) has positive Lebesgue measure."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.15328/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"}