{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:RMNJYSTPJKVGR3NKSMZTF456UC","short_pith_number":"pith:RMNJYSTP","canonical_record":{"source":{"id":"1705.03871","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-10T17:46:44Z","cross_cats_sorted":[],"title_canon_sha256":"3c7e1f3dfb219b58bdebbd0269d595a1fc77cb6c8040345b44c79fa4a68cb093","abstract_canon_sha256":"27e046d5584c2d33c34a6e64965abb9c168cc3709a7133baf0141f4502724ce6"},"schema_version":"1.0"},"canonical_sha256":"8b1a9c4a6f4aaa68edaa933332f3bea09161a5496d3cb833ae867f952181f86c","source":{"kind":"arxiv","id":"1705.03871","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.03871","created_at":"2026-05-18T00:44:42Z"},{"alias_kind":"arxiv_version","alias_value":"1705.03871v1","created_at":"2026-05-18T00:44:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.03871","created_at":"2026-05-18T00:44:42Z"},{"alias_kind":"pith_short_12","alias_value":"RMNJYSTPJKVG","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"RMNJYSTPJKVGR3NK","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"RMNJYSTP","created_at":"2026-05-18T12:31:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:RMNJYSTPJKVGR3NKSMZTF456UC","target":"record","payload":{"canonical_record":{"source":{"id":"1705.03871","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-10T17:46:44Z","cross_cats_sorted":[],"title_canon_sha256":"3c7e1f3dfb219b58bdebbd0269d595a1fc77cb6c8040345b44c79fa4a68cb093","abstract_canon_sha256":"27e046d5584c2d33c34a6e64965abb9c168cc3709a7133baf0141f4502724ce6"},"schema_version":"1.0"},"canonical_sha256":"8b1a9c4a6f4aaa68edaa933332f3bea09161a5496d3cb833ae867f952181f86c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:42.890867Z","signature_b64":"+eoJEoZup+HIqTzd/Vd6Kp5F09RNQ9H1gwu8OYOtLGO+n2UB8ILYoZnW1Oege0JhikJPdLgHmewVABk7zy3JCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b1a9c4a6f4aaa68edaa933332f3bea09161a5496d3cb833ae867f952181f86c","last_reissued_at":"2026-05-18T00:44:42.890404Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:42.890404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.03871","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:44:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DPNi1lsx+U8AgcKta9zvr/TBmwX9wW7cJ7mzXrpVSFKhstJ78ejrx+s0hBsFUK+Zj52KocI/hwWL6FjBRc/6Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T17:59:08.074019Z"},"content_sha256":"4f6c89a80a16c27a9719d8af0b8bab41f10db69e71eaac2a792925f072ca0ae9","schema_version":"1.0","event_id":"sha256:4f6c89a80a16c27a9719d8af0b8bab41f10db69e71eaac2a792925f072ca0ae9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:RMNJYSTPJKVGR3NKSMZTF456UC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Group actions and a multi-parameter Falconer distance problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Alex Rice, Kyle Hambrook","submitted_at":"2017-05-10T17:46:44Z","abstract_excerpt":"In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given ${\\textbf{d}}=(d_1,d_2, \\dots, d_{\\ell})\\in \\mathbb{N}^{\\ell}$ with $d_1+d_2+\\dots+d_{\\ell}=d$ and $E \\subseteq \\mathbb{R}^d$, we define $$ \\Delta_{{\\textbf{d}}}(E) = \\left\\{ \\left(|x^{(1)}-y^{(1)}|,\\ldots,|x^{(\\ell)}-y^{(\\ell)}|\\right) : x,y \\in E \\right\\} \\subseteq \\mathbb{R}^{\\ell}, $$ where for $x\\in \\mathbb{R}^d$ we write $x=\\left( x^{(1)},\\dots, x^{(\\ell)} \\right)$ with $x^{(i)} \\in \\mathbb{R}^{d_i}$.\n  We ask how large does the Hausdorff dimension of $E$ need to be to ensure t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.03871","kind":"arxiv","version":1},"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":"2026-05-18T00:44:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A+etEMifBHQOM/jllCY+OQcqBgm7JPpypuX4iBxf1A1mHsGIP2Rc36xe7k8+tHtLsL//bDwkbjjmSKsTr7/QCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T17:59:08.074654Z"},"content_sha256":"78cf340d0ecc7e0f6d9e6647399be10721eb86121f3abe73ae0ff12b13d4237f","schema_version":"1.0","event_id":"sha256:78cf340d0ecc7e0f6d9e6647399be10721eb86121f3abe73ae0ff12b13d4237f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/bundle.json","state_url":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RMNJYSTPJKVGR3NKSMZTF456UC/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":"2026-05-23T17:59:08Z","links":{"resolver":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC","bundle":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/bundle.json","state":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RMNJYSTPJKVGR3NKSMZTF456UC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:RMNJYSTPJKVGR3NKSMZTF456UC","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":"27e046d5584c2d33c34a6e64965abb9c168cc3709a7133baf0141f4502724ce6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-10T17:46:44Z","title_canon_sha256":"3c7e1f3dfb219b58bdebbd0269d595a1fc77cb6c8040345b44c79fa4a68cb093"},"schema_version":"1.0","source":{"id":"1705.03871","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.03871","created_at":"2026-05-18T00:44:42Z"},{"alias_kind":"arxiv_version","alias_value":"1705.03871v1","created_at":"2026-05-18T00:44:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.03871","created_at":"2026-05-18T00:44:42Z"},{"alias_kind":"pith_short_12","alias_value":"RMNJYSTPJKVG","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"RMNJYSTPJKVGR3NK","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"RMNJYSTP","created_at":"2026-05-18T12:31:39Z"}],"graph_snapshots":[{"event_id":"sha256:78cf340d0ecc7e0f6d9e6647399be10721eb86121f3abe73ae0ff12b13d4237f","target":"graph","created_at":"2026-05-18T00:44:42Z","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":"In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given ${\\textbf{d}}=(d_1,d_2, \\dots, d_{\\ell})\\in \\mathbb{N}^{\\ell}$ with $d_1+d_2+\\dots+d_{\\ell}=d$ and $E \\subseteq \\mathbb{R}^d$, we define $$ \\Delta_{{\\textbf{d}}}(E) = \\left\\{ \\left(|x^{(1)}-y^{(1)}|,\\ldots,|x^{(\\ell)}-y^{(\\ell)}|\\right) : x,y \\in E \\right\\} \\subseteq \\mathbb{R}^{\\ell}, $$ where for $x\\in \\mathbb{R}^d$ we write $x=\\left( x^{(1)},\\dots, x^{(\\ell)} \\right)$ with $x^{(i)} \\in \\mathbb{R}^{d_i}$.\n  We ask how large does the Hausdorff dimension of $E$ need to be to ensure t","authors_text":"Alex Iosevich, Alex Rice, Kyle Hambrook","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-10T17:46:44Z","title":"Group actions and a multi-parameter Falconer distance problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.03871","kind":"arxiv","version":1},"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":"sha256:4f6c89a80a16c27a9719d8af0b8bab41f10db69e71eaac2a792925f072ca0ae9","target":"record","created_at":"2026-05-18T00:44:42Z","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":"27e046d5584c2d33c34a6e64965abb9c168cc3709a7133baf0141f4502724ce6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-10T17:46:44Z","title_canon_sha256":"3c7e1f3dfb219b58bdebbd0269d595a1fc77cb6c8040345b44c79fa4a68cb093"},"schema_version":"1.0","source":{"id":"1705.03871","kind":"arxiv","version":1}},"canonical_sha256":"8b1a9c4a6f4aaa68edaa933332f3bea09161a5496d3cb833ae867f952181f86c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8b1a9c4a6f4aaa68edaa933332f3bea09161a5496d3cb833ae867f952181f86c","first_computed_at":"2026-05-18T00:44:42.890404Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:42.890404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+eoJEoZup+HIqTzd/Vd6Kp5F09RNQ9H1gwu8OYOtLGO+n2UB8ILYoZnW1Oege0JhikJPdLgHmewVABk7zy3JCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:42.890867Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.03871","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f6c89a80a16c27a9719d8af0b8bab41f10db69e71eaac2a792925f072ca0ae9","sha256:78cf340d0ecc7e0f6d9e6647399be10721eb86121f3abe73ae0ff12b13d4237f"],"state_sha256":"228faf9a71d06a6d6dd777a1ebb618342002a098a1743976232192e551ed4b50"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w8kGWrfDHjpP2H8AVsx2xv7ELdo1KOosml0d2MJQgN89NcVF0epZ2WxuaIjuzZwftMaDADeGyrJrgB7Gj3q/CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T17:59:08.078174Z","bundle_sha256":"48dacd0a211d293f18824e1006fadb5a1513e0d0e447db318eceefa84eeddd41"}}