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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} $g$ ","authors_text":"Eric Schmutz, Michael Tait","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-01-20T20:41:03Z","title":"Cardinalities of $g$-difference sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.11736","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":"sha256:113337549cc38cc6c686b7532618355206a0aec48594361f2547dd591906831d","target":"record","created_at":"2026-07-05T10:03:13Z","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":"7d09e16310fae474084d7ed16f7efd77273e9afc2deb8d4f418ee705fcf26cf5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-01-20T20:41:03Z","title_canon_sha256":"e35ec648630876246277f6bbeb9bf7d4fd41b0760e2326754cfa6e4926300322"},"schema_version":"1.0","source":{"id":"2501.11736","kind":"arxiv","version":1}},"canonical_sha256":"9ea020b5c12d25cbd5e73087407fc24c1c8254fe7dcb8411ba1e4400684ce46e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ea020b5c12d25cbd5e73087407fc24c1c8254fe7dcb8411ba1e4400684ce46e","first_computed_at":"2026-07-05T10:03:13.195979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:03:13.195979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n6m7EoDIXFFF43p/coHpimVUnckFtvHiZ7SXGQkww9WwyxWBqskOb9PrfZqltwzrtkd41mhcNmOiRWhtq2qCAg==","signature_status":"signed_v1","signed_at":"2026-07-05T10:03:13.196454Z","signed_message":"canonical_sha256_bytes"},"source_id":"2501.11736","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:113337549cc38cc6c686b7532618355206a0aec48594361f2547dd591906831d","sha256:d309dbf0b2792137cbec8f43399f0dd41d4af58ea088c3bc185be80dd987d1fd"],"state_sha256":"d1a8e35d50877b1d17ff8835af2101aaa72bbf5477134d6bd6dde0eec589d173"}