{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:BP2LFJO5D2AH6ZZJ4KE6XCBCHW","short_pith_number":"pith:BP2LFJO5","canonical_record":{"source":{"id":"1009.2471","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-09-13T18:27:38Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"cdcdb58cf7fb66600b377d942071a417c8e3172674982914910bd05352ae378c","abstract_canon_sha256":"8f2e81989663fbd3ff65a8d9489e910b922007cba6d412a372e306192ee7594a"},"schema_version":"1.0"},"canonical_sha256":"0bf4b2a5dd1e807f6729e289eb88223d91c9e10133d807a7d7cbbd17c0084973","source":{"kind":"arxiv","id":"1009.2471","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2471","created_at":"2026-05-18T04:09:45Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2471v4","created_at":"2026-05-18T04:09:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2471","created_at":"2026-05-18T04:09:45Z"},{"alias_kind":"pith_short_12","alias_value":"BP2LFJO5D2AH","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"BP2LFJO5D2AH6ZZJ","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"BP2LFJO5","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:BP2LFJO5D2AH6ZZJ4KE6XCBCHW","target":"record","payload":{"canonical_record":{"source":{"id":"1009.2471","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-09-13T18:27:38Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"cdcdb58cf7fb66600b377d942071a417c8e3172674982914910bd05352ae378c","abstract_canon_sha256":"8f2e81989663fbd3ff65a8d9489e910b922007cba6d412a372e306192ee7594a"},"schema_version":"1.0"},"canonical_sha256":"0bf4b2a5dd1e807f6729e289eb88223d91c9e10133d807a7d7cbbd17c0084973","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:45.870956Z","signature_b64":"/ClfnuQA7Y1zS7T/u/hLBiwHACeoiHgufRaK8Mdc/P9szspkhedb4hdv7e7++l1xmN8JhashjHXyW810gWyGBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0bf4b2a5dd1e807f6729e289eb88223d91c9e10133d807a7d7cbbd17c0084973","last_reissued_at":"2026-05-18T04:09:45.870541Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:45.870541Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.2471","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:09:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u31pW+IvSq0haKNENH4lEVLbyVrrV96pxc8sU0qA31bO9X2EWwjV39rL6nFvrilZf9JReHdT/KmI9paL9tnOCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:12:13.434983Z"},"content_sha256":"1417d3a9bf245919b41211c2392f6ebca0c59db20aaeacc3395e3126c23cf54d","schema_version":"1.0","event_id":"sha256:1417d3a9bf245919b41211c2392f6ebca0c59db20aaeacc3395e3126c23cf54d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:BP2LFJO5D2AH6ZZJ4KE6XCBCHW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Allan Greenleaf","submitted_at":"2010-09-13T18:27:38Z","abstract_excerpt":"We prove that if the Hausdorff dimension of a compact set $E \\subset {\\Bbb R}^2$ is greater than 7/4, then the set of {\\ag three-point configurations determined by $E$ has positive three-dimensional measure}. We establish this by showing that {\\ag a} natural measure on the set of {\\ag such configurations} has {\\ag Radon-Nikodym derivative} in $L^{\\infty}$ if $\\dH(E)> 7/4$, and the index 7/4 in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator, $$ B(f,g)(x)=\\int \\int f(x-u) g(x-v)\\, dK(u,v),$$ where $K$ is surface meas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2471","kind":"arxiv","version":4},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:09:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l+8kI2dd8PErFr9ZitGM3afcYCp7XKaoBxOMpdcAAZgwfue5fkQyE9KhSmvrG3d+K0BYdw7ZTP5T6ZkiLvFMBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:12:13.435352Z"},"content_sha256":"1e24012cae948f54616a79834f07b56bbcb32674d7992a9368814e74ef36b491","schema_version":"1.0","event_id":"sha256:1e24012cae948f54616a79834f07b56bbcb32674d7992a9368814e74ef36b491"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BP2LFJO5D2AH6ZZJ4KE6XCBCHW/bundle.json","state_url":"https://pith.science/pith/BP2LFJO5D2AH6ZZJ4KE6XCBCHW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BP2LFJO5D2AH6ZZJ4KE6XCBCHW/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T16:12:13Z","links":{"resolver":"https://pith.science/pith/BP2LFJO5D2AH6ZZJ4KE6XCBCHW","bundle":"https://pith.science/pith/BP2LFJO5D2AH6ZZJ4KE6XCBCHW/bundle.json","state":"https://pith.science/pith/BP2LFJO5D2AH6ZZJ4KE6XCBCHW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BP2LFJO5D2AH6ZZJ4KE6XCBCHW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:BP2LFJO5D2AH6ZZJ4KE6XCBCHW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8f2e81989663fbd3ff65a8d9489e910b922007cba6d412a372e306192ee7594a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-09-13T18:27:38Z","title_canon_sha256":"cdcdb58cf7fb66600b377d942071a417c8e3172674982914910bd05352ae378c"},"schema_version":"1.0","source":{"id":"1009.2471","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2471","created_at":"2026-05-18T04:09:45Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2471v4","created_at":"2026-05-18T04:09:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2471","created_at":"2026-05-18T04:09:45Z"},{"alias_kind":"pith_short_12","alias_value":"BP2LFJO5D2AH","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"BP2LFJO5D2AH6ZZJ","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"BP2LFJO5","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:1e24012cae948f54616a79834f07b56bbcb32674d7992a9368814e74ef36b491","target":"graph","created_at":"2026-05-18T04:09:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove that if the Hausdorff dimension of a compact set $E \\subset {\\Bbb R}^2$ is greater than 7/4, then the set of {\\ag three-point configurations determined by $E$ has positive three-dimensional measure}. We establish this by showing that {\\ag a} natural measure on the set of {\\ag such configurations} has {\\ag Radon-Nikodym derivative} in $L^{\\infty}$ if $\\dH(E)> 7/4$, and the index 7/4 in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator, $$ B(f,g)(x)=\\int \\int f(x-u) g(x-v)\\, dK(u,v),$$ where $K$ is surface meas","authors_text":"Alex Iosevich, Allan Greenleaf","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-09-13T18:27:38Z","title":"On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2471","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1417d3a9bf245919b41211c2392f6ebca0c59db20aaeacc3395e3126c23cf54d","target":"record","created_at":"2026-05-18T04:09:45Z","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":"8f2e81989663fbd3ff65a8d9489e910b922007cba6d412a372e306192ee7594a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-09-13T18:27:38Z","title_canon_sha256":"cdcdb58cf7fb66600b377d942071a417c8e3172674982914910bd05352ae378c"},"schema_version":"1.0","source":{"id":"1009.2471","kind":"arxiv","version":4}},"canonical_sha256":"0bf4b2a5dd1e807f6729e289eb88223d91c9e10133d807a7d7cbbd17c0084973","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0bf4b2a5dd1e807f6729e289eb88223d91c9e10133d807a7d7cbbd17c0084973","first_computed_at":"2026-05-18T04:09:45.870541Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:45.870541Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/ClfnuQA7Y1zS7T/u/hLBiwHACeoiHgufRaK8Mdc/P9szspkhedb4hdv7e7++l1xmN8JhashjHXyW810gWyGBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:45.870956Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.2471","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1417d3a9bf245919b41211c2392f6ebca0c59db20aaeacc3395e3126c23cf54d","sha256:1e24012cae948f54616a79834f07b56bbcb32674d7992a9368814e74ef36b491"],"state_sha256":"653ca9c011427f249e15aa45463cc378b469289024b340bbe27db37f1860e3a4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oBRfBkukOzC+vwHzHGPS+SkEvqlPVhheDLUAG+PRA7PYry6OSJb8FQ7KCE3//5V0rtXA6pu94ZOS73Y+aBSsDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T16:12:13.437311Z","bundle_sha256":"13c3c64a5ce9652745ce0ceea6fb92661323ce98148d0fecb755bcccc9cab682"}}