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Given $E$ and $F$ a point sets in $\\mathbb{R}^d$ and $p = (p_1, \\ldots, p_q)$ with $p_1+ \\cdots + p_q = d$ is an increasing partition of $d$ define $$ B_p(E,F)=\\{(|x_1-y_1|, \\ldots, |x_q-y_q|): x \\in E, y \\in F \\},$$ where $x=(x_1, \\ldots, x_q)$ with $x_i$ in $\\mathbb{R}^{p_i}$. For $p_1 \\geq 2$ it is not difficult to construct $E$ and $F$ such that $|B_{p}(E,F)|=1$. On the other hand, it is easy to see that if $\\gamma_q$ is the best know exponent for the distance problem in $\\mathbb{R}^{p_i}$ that $|B_p(E,E)| \\geq C{|E|}^{\\frac"},"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":"1712.04060","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-11T22:49:07Z","cross_cats_sorted":[],"title_canon_sha256":"4bdd8c8e1963efd6197337deee2c3cdad8bac23f3b823dc92659ba3ef04e0e3e","abstract_canon_sha256":"2ca0002139b10f7af399a57bc000a3774bf5ffb2a33c07f1d8b8c538a084733c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:09.348984Z","signature_b64":"c+IdYGL8kOVIdZxXLuXmJpda5teA9l2/PpTO3sqly4KLMUh8d/v3zKDWT5PZQYgYxbemmFV43y3jzcu291ybDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9988dc772fa37b128e8ed1016c7f50f78dd35acfd0c64df666e6309ca565e3dc","last_reissued_at":"2026-05-18T00:28:09.348219Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:09.348219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A multi-parameter variant of the Erd\\H{o}s distance problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Iosevich, Jonathan Passant, Maria Janczak","submitted_at":"2017-12-11T22:49:07Z","abstract_excerpt":"We study the following variant of the Erd\\H{o}s distance problem. Given $E$ and $F$ a point sets in $\\mathbb{R}^d$ and $p = (p_1, \\ldots, p_q)$ with $p_1+ \\cdots + p_q = d$ is an increasing partition of $d$ define $$ B_p(E,F)=\\{(|x_1-y_1|, \\ldots, |x_q-y_q|): x \\in E, y \\in F \\},$$ where $x=(x_1, \\ldots, x_q)$ with $x_i$ in $\\mathbb{R}^{p_i}$. For $p_1 \\geq 2$ it is not difficult to construct $E$ and $F$ such that $|B_{p}(E,F)|=1$. On the other hand, it is easy to see that if $\\gamma_q$ is the best know exponent for the distance problem in $\\mathbb{R}^{p_i}$ that $|B_p(E,E)| \\geq C{|E|}^{\\frac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04060","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":"1712.04060","created_at":"2026-05-18T00:28:09.348340+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.04060v1","created_at":"2026-05-18T00:28:09.348340+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.04060","created_at":"2026-05-18T00:28:09.348340+00:00"},{"alias_kind":"pith_short_12","alias_value":"TGENY5ZPUN5R","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_16","alias_value":"TGENY5ZPUN5RFDUO","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_8","alias_value":"TGENY5ZP","created_at":"2026-05-18T12:31:46.661854+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/TGENY5ZPUN5RFDUO2EAWY72Q66","json":"https://pith.science/pith/TGENY5ZPUN5RFDUO2EAWY72Q66.json","graph_json":"https://pith.science/api/pith-number/TGENY5ZPUN5RFDUO2EAWY72Q66/graph.json","events_json":"https://pith.science/api/pith-number/TGENY5ZPUN5RFDUO2EAWY72Q66/events.json","paper":"https://pith.science/paper/TGENY5ZP"},"agent_actions":{"view_html":"https://pith.science/pith/TGENY5ZPUN5RFDUO2EAWY72Q66","download_json":"https://pith.science/pith/TGENY5ZPUN5RFDUO2EAWY72Q66.json","view_paper":"https://pith.science/paper/TGENY5ZP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.04060&json=true","fetch_graph":"https://pith.science/api/pith-number/TGENY5ZPUN5RFDUO2EAWY72Q66/graph.json","fetch_events":"https://pith.science/api/pith-number/TGENY5ZPUN5RFDUO2EAWY72Q66/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TGENY5ZPUN5RFDUO2EAWY72Q66/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TGENY5ZPUN5RFDUO2EAWY72Q66/action/storage_attestation","attest_author":"https://pith.science/pith/TGENY5ZPUN5RFDUO2EAWY72Q66/action/author_attestation","sign_citation":"https://pith.science/pith/TGENY5ZPUN5RFDUO2EAWY72Q66/action/citation_signature","submit_replication":"https://pith.science/pith/TGENY5ZPUN5RFDUO2EAWY72Q66/action/replication_record"}},"created_at":"2026-05-18T00:28:09.348340+00:00","updated_at":"2026-05-18T00:28:09.348340+00:00"}