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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1705.03871","created_at":"2026-05-18T00:44:42.890473+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.03871v1","created_at":"2026-05-18T00:44:42.890473+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.03871","created_at":"2026-05-18T00:44:42.890473+00:00"},{"alias_kind":"pith_short_12","alias_value":"RMNJYSTPJKVG","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_16","alias_value":"RMNJYSTPJKVGR3NK","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_8","alias_value":"RMNJYSTP","created_at":"2026-05-18T12:31:39.905425+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC","json":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC.json","graph_json":"https://pith.science/api/pith-number/RMNJYSTPJKVGR3NKSMZTF456UC/graph.json","events_json":"https://pith.science/api/pith-number/RMNJYSTPJKVGR3NKSMZTF456UC/events.json","paper":"https://pith.science/paper/RMNJYSTP"},"agent_actions":{"view_html":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC","download_json":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC.json","view_paper":"https://pith.science/paper/RMNJYSTP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.03871&json=true","fetch_graph":"https://pith.science/api/pith-number/RMNJYSTPJKVGR3NKSMZTF456UC/graph.json","fetch_events":"https://pith.science/api/pith-number/RMNJYSTPJKVGR3NKSMZTF456UC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/action/storage_attestation","attest_author":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/action/author_attestation","sign_citation":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/action/citation_signature","submit_replication":"https://pith.science/pith/RMNJYSTPJKVGR3NKSMZTF456UC/action/replication_record"}},"created_at":"2026-05-18T00:44:42.890473+00:00","updated_at":"2026-05-18T00:44:42.890473+00:00"}