{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:WI4XC2722KWACC3GOJOA3XG7XD","short_pith_number":"pith:WI4XC272","schema_version":"1.0","canonical_sha256":"b239716bfad2ac010b66725c0ddcdfb8cf7dae57e03e14b5c09be24f42f4fda1","source":{"kind":"arxiv","id":"1208.1413","version":1},"attestation_state":"computed","paper":{"title":"Optimal number representations in negative base","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Edita Pelantov\\'a, Zuzana Mas\\'akov\\'a","submitted_at":"2012-08-07T13:08:20Z","abstract_excerpt":"For a given base $\\gamma$ and a digit set ${\\mathcal B}$ we consider optimal representations of a number $x$, as defined by Dajani at al. in 2012. For a non-integer negative base $\\gamma=-\\beta<-1$ and the digit set ${\\mathcal A}_\\beta:={0,1,...,\\lceil\\beta\\rceil-1}$ we derive the transformation which generates the optimal representation, if it exists. We show that -- unlike the case of negative integer base -- almost no $x$ has an optimal representation. For a positive base $\\gamma=\\beta>1$ and the alphabet ${\\mathcal A}_\\beta$ we provide an alternative proof of statements obtained by Dajani "},"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":"1208.1413","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-08-07T13:08:20Z","cross_cats_sorted":[],"title_canon_sha256":"f47e41b7711966958917dc7981bd051dac0f29b25b53aa2f909399f7f8161101","abstract_canon_sha256":"991055b81a1b35ab16afe44d7ec339403f9f563fd7717af69029585b1a017988"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:12.522138Z","signature_b64":"KpiaHq/n4H4MaCQ7X+LjcbSdbNq2TQJthHbI5lN3qIhPZWpK4uETL6fI6Ea23FVE6Zx10iQRkjO903q0UwU8DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b239716bfad2ac010b66725c0ddcdfb8cf7dae57e03e14b5c09be24f42f4fda1","last_reissued_at":"2026-05-18T03:49:12.521342Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:12.521342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal number representations in negative base","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Edita Pelantov\\'a, Zuzana Mas\\'akov\\'a","submitted_at":"2012-08-07T13:08:20Z","abstract_excerpt":"For a given base $\\gamma$ and a digit set ${\\mathcal B}$ we consider optimal representations of a number $x$, as defined by Dajani at al. in 2012. For a non-integer negative base $\\gamma=-\\beta<-1$ and the digit set ${\\mathcal A}_\\beta:={0,1,...,\\lceil\\beta\\rceil-1}$ we derive the transformation which generates the optimal representation, if it exists. We show that -- unlike the case of negative integer base -- almost no $x$ has an optimal representation. For a positive base $\\gamma=\\beta>1$ and the alphabet ${\\mathcal A}_\\beta$ we provide an alternative proof of statements obtained by Dajani "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1413","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":"1208.1413","created_at":"2026-05-18T03:49:12.521472+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.1413v1","created_at":"2026-05-18T03:49:12.521472+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.1413","created_at":"2026-05-18T03:49:12.521472+00:00"},{"alias_kind":"pith_short_12","alias_value":"WI4XC2722KWA","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"WI4XC2722KWACC3G","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"WI4XC272","created_at":"2026-05-18T12:27:25.539911+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/WI4XC2722KWACC3GOJOA3XG7XD","json":"https://pith.science/pith/WI4XC2722KWACC3GOJOA3XG7XD.json","graph_json":"https://pith.science/api/pith-number/WI4XC2722KWACC3GOJOA3XG7XD/graph.json","events_json":"https://pith.science/api/pith-number/WI4XC2722KWACC3GOJOA3XG7XD/events.json","paper":"https://pith.science/paper/WI4XC272"},"agent_actions":{"view_html":"https://pith.science/pith/WI4XC2722KWACC3GOJOA3XG7XD","download_json":"https://pith.science/pith/WI4XC2722KWACC3GOJOA3XG7XD.json","view_paper":"https://pith.science/paper/WI4XC272","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.1413&json=true","fetch_graph":"https://pith.science/api/pith-number/WI4XC2722KWACC3GOJOA3XG7XD/graph.json","fetch_events":"https://pith.science/api/pith-number/WI4XC2722KWACC3GOJOA3XG7XD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WI4XC2722KWACC3GOJOA3XG7XD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WI4XC2722KWACC3GOJOA3XG7XD/action/storage_attestation","attest_author":"https://pith.science/pith/WI4XC2722KWACC3GOJOA3XG7XD/action/author_attestation","sign_citation":"https://pith.science/pith/WI4XC2722KWACC3GOJOA3XG7XD/action/citation_signature","submit_replication":"https://pith.science/pith/WI4XC2722KWACC3GOJOA3XG7XD/action/replication_record"}},"created_at":"2026-05-18T03:49:12.521472+00:00","updated_at":"2026-05-18T03:49:12.521472+00:00"}