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We prove that if $A$ is a $\\gamma$-Golomb ruler, then \\[\\liminf_{n\\to\\infty} \\frac{|A\\cap[0,n)|}{\\sqrt{n/\\log n}} \\le \\frac{2}{\\sqrt{\\log 2}} \\sqrt{\\gamma}.\\] There is a $\\gamma$-Golomb ruler $G$ with \\[ \\limsup_{n\\to\\infty} \\frac{|G\\cap[0,n)|}{\\sqrt n} \\ge \\frac{1}{\\sqrt2} \\sqrt{\\gamma}.\\]"},"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":"2606.28651","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-26T23:34:53Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"8b7772d2977dfb7e885028c0c7ef1c8a4b64540c3fa59917af2fc37cbc20b96e","abstract_canon_sha256":"ed54fc637be2ea876c426e6dc3d927fe919881be92eff4ef4def4fb44066c9ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:16:45.922170Z","signature_b64":"L/E9FrUGsTemuZqbrmNuERfU2huAOSEpVw3M9NuuYkJfnJFVqY5l9DR31yH/q3mpZ1/KdBj+l2n82G6kp3rDCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b54ed677fc9986bc9ddbc0e0b4a0ab5d7e5b3e91aee34e2cb48dacae29e9b84","last_reissued_at":"2026-06-30T01:16:45.921513Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:16:45.921513Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Thickness of Infinite Sidon Sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Kevin O'Bryant","submitted_at":"2026-06-26T23:34:53Z","abstract_excerpt":"Let $\\gamma \\ge 1$. A set $A$ of nonnegative integers is a Sidon set if for each $d>0$ there is at most one pair $(a,b) \\in A \\times A$ with $d=a-b$. If there are at most $\\gamma$ pairs, then $A$ is a $\\gamma$-Golomb ruler. We prove that if $A$ is a $\\gamma$-Golomb ruler, then \\[\\liminf_{n\\to\\infty} \\frac{|A\\cap[0,n)|}{\\sqrt{n/\\log n}} \\le \\frac{2}{\\sqrt{\\log 2}} \\sqrt{\\gamma}.\\] There is a $\\gamma$-Golomb ruler $G$ with \\[ \\limsup_{n\\to\\infty} \\frac{|G\\cap[0,n)|}{\\sqrt n} \\ge \\frac{1}{\\sqrt2} \\sqrt{\\gamma}.\\]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28651","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/2606.28651/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":"2606.28651","created_at":"2026-06-30T01:16:45.921586+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.28651v1","created_at":"2026-06-30T01:16:45.921586+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.28651","created_at":"2026-06-30T01:16:45.921586+00:00"},{"alias_kind":"pith_short_12","alias_value":"NNKO2Z37ZGMG","created_at":"2026-06-30T01:16:45.921586+00:00"},{"alias_kind":"pith_short_16","alias_value":"NNKO2Z37ZGMGXSO5","created_at":"2026-06-30T01:16:45.921586+00:00"},{"alias_kind":"pith_short_8","alias_value":"NNKO2Z37","created_at":"2026-06-30T01:16:45.921586+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/NNKO2Z37ZGMGXSO5XQHAWSQKWX","json":"https://pith.science/pith/NNKO2Z37ZGMGXSO5XQHAWSQKWX.json","graph_json":"https://pith.science/api/pith-number/NNKO2Z37ZGMGXSO5XQHAWSQKWX/graph.json","events_json":"https://pith.science/api/pith-number/NNKO2Z37ZGMGXSO5XQHAWSQKWX/events.json","paper":"https://pith.science/paper/NNKO2Z37"},"agent_actions":{"view_html":"https://pith.science/pith/NNKO2Z37ZGMGXSO5XQHAWSQKWX","download_json":"https://pith.science/pith/NNKO2Z37ZGMGXSO5XQHAWSQKWX.json","view_paper":"https://pith.science/paper/NNKO2Z37","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.28651&json=true","fetch_graph":"https://pith.science/api/pith-number/NNKO2Z37ZGMGXSO5XQHAWSQKWX/graph.json","fetch_events":"https://pith.science/api/pith-number/NNKO2Z37ZGMGXSO5XQHAWSQKWX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NNKO2Z37ZGMGXSO5XQHAWSQKWX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NNKO2Z37ZGMGXSO5XQHAWSQKWX/action/storage_attestation","attest_author":"https://pith.science/pith/NNKO2Z37ZGMGXSO5XQHAWSQKWX/action/author_attestation","sign_citation":"https://pith.science/pith/NNKO2Z37ZGMGXSO5XQHAWSQKWX/action/citation_signature","submit_replication":"https://pith.science/pith/NNKO2Z37ZGMGXSO5XQHAWSQKWX/action/replication_record"}},"created_at":"2026-06-30T01:16:45.921586+00:00","updated_at":"2026-06-30T01:16:45.921586+00:00"}