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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.","authors_text":"Bochen Liu","cross_cats":["math.AP","math.CO"],"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.","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CA","submitted_at":"2026-03-16T14:20:56Z","title":"Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.15328","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-15T10:25:21.289653Z","id":"4973fefb-539f-413b-9a61-5cf109904fcc","model_set":{"reader":"grok-4.3"},"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","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.","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.","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."}},"verdict_id":"4973fefb-539f-413b-9a61-5cf109904fcc"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:833596ff4115c8fd922192dd5e33707b2354b1cb94b22f6a26e4ce4f9a0906dc","target":"record","created_at":"2026-07-17T01:20:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f6fad89910c6a008ebe7ba6b57f52a48aeb97f85e639e189be40b77969ae8076","cross_cats_sorted":["math.AP","math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CA","submitted_at":"2026-03-16T14:20:56Z","title_canon_sha256":"aaeac579701ef0901db5c42e7ddcc17f7745974c1258dc414029b64a2387e36c"},"schema_version":"1.0","source":{"id":"2603.15328","kind":"arxiv","version":3}},"canonical_sha256":"867958de48003929d1ce91035addaae2c6087fdf775af57f7d8f9c31367db6d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"867958de48003929d1ce91035addaae2c6087fdf775af57f7d8f9c31367db6d0","first_computed_at":"2026-07-17T01:20:47.501936Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-17T01:20:47.501936Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MlZztDcFIAdlvkSUR93vYvvwxiQD51GTBEqj0xp9YcQWFhjcYcvwTpnbAEcS3MJ56nouFOgAMRCxM3zWdfaZAg==","signature_status":"signed_v1","signed_at":"2026-07-17T01:20:47.502918Z","signed_message":"canonical_sha256_bytes"},"source_id":"2603.15328","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:833596ff4115c8fd922192dd5e33707b2354b1cb94b22f6a26e4ce4f9a0906dc","sha256:b930cf81f276b5e9b9d8c3b0e2069578fbe07ad14d836c6f72b32bf83b82e022"],"state_sha256":"53da9b47fb93d323d4774f24244c291952f4d65c5721409de41b4908cba1e930"}