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We prove that the finite, non-zero limit $\\lim\\limits_{n\\rightarrow \\infty}\\frac{\\eta_{g}(n)}{\\sqrt{n}}$ exists, answering a question of Kravitz. We also investigate a similar problem in the setting of a vector space over a finite field.\n  Let $\\alpha_g(n)$ be the largest cardinality that $A\\subseteq [n]$ can have if, for all nonzero $x$, $a_{1}-a_{2}=x$ has {\\em at most} $g$ "},"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":"2501.11736","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-01-20T20:41:03Z","cross_cats_sorted":[],"title_canon_sha256":"e35ec648630876246277f6bbeb9bf7d4fd41b0760e2326754cfa6e4926300322","abstract_canon_sha256":"7d09e16310fae474084d7ed16f7efd77273e9afc2deb8d4f418ee705fcf26cf5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:03:13.196454Z","signature_b64":"n6m7EoDIXFFF43p/coHpimVUnckFtvHiZ7SXGQkww9WwyxWBqskOb9PrfZqltwzrtkd41mhcNmOiRWhtq2qCAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ea020b5c12d25cbd5e73087407fc24c1c8254fe7dcb8411ba1e4400684ce46e","last_reissued_at":"2026-07-05T10:03:13.195979Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:03:13.195979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cardinalities of $g$-difference sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eric Schmutz, Michael Tait","submitted_at":"2025-01-20T20:41:03Z","abstract_excerpt":"Let $\\eta_{g}(n) $ be the smallest cardinality that $A\\subseteq {\\mathbb Z}$ can have if $A$ is a $g$-difference basis for $[n]$ (i.e, if, for each $x\\in [n]$, there are {\\em at least} $g$ solutions to $a_{1}-a_{2}=x$ ). We prove that the finite, non-zero limit $\\lim\\limits_{n\\rightarrow \\infty}\\frac{\\eta_{g}(n)}{\\sqrt{n}}$ exists, answering a question of Kravitz. We also investigate a similar problem in the setting of a vector space over a finite field.\n  Let $\\alpha_g(n)$ be the largest cardinality that $A\\subseteq [n]$ can have if, for all nonzero $x$, $a_{1}-a_{2}=x$ has {\\em at most} $g$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.11736","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.11736/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2501.11736","created_at":"2026-07-05T10:03:13.196036+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.11736v1","created_at":"2026-07-05T10:03:13.196036+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.11736","created_at":"2026-07-05T10:03:13.196036+00:00"},{"alias_kind":"pith_short_12","alias_value":"T2QCBNOBFUS4","created_at":"2026-07-05T10:03:13.196036+00:00"},{"alias_kind":"pith_short_16","alias_value":"T2QCBNOBFUS4XVPH","created_at":"2026-07-05T10:03:13.196036+00:00"},{"alias_kind":"pith_short_8","alias_value":"T2QCBNOB","created_at":"2026-07-05T10:03:13.196036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.19833","citing_title":"Fault-Tolerant Shared-Relay Communication in Circulant Interconnection Networks","ref_index":14,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ","json":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ.json","graph_json":"https://pith.science/api/pith-number/T2QCBNOBFUS4XVPHGCDUA76CJQ/graph.json","events_json":"https://pith.science/api/pith-number/T2QCBNOBFUS4XVPHGCDUA76CJQ/events.json","paper":"https://pith.science/paper/T2QCBNOB"},"agent_actions":{"view_html":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ","download_json":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ.json","view_paper":"https://pith.science/paper/T2QCBNOB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.11736&json=true","fetch_graph":"https://pith.science/api/pith-number/T2QCBNOBFUS4XVPHGCDUA76CJQ/graph.json","fetch_events":"https://pith.science/api/pith-number/T2QCBNOBFUS4XVPHGCDUA76CJQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ/action/storage_attestation","attest_author":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ/action/author_attestation","sign_citation":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ/action/citation_signature","submit_replication":"https://pith.science/pith/T2QCBNOBFUS4XVPHGCDUA76CJQ/action/replication_record"}},"created_at":"2026-07-05T10:03:13.196036+00:00","updated_at":"2026-07-05T10:03:13.196036+00:00"}