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Pisot numbers play a crucial role for the properties of $X^m(\\beta)$. Following the work of Za\\\"imi, who considered $X^m(\\gamma)$ with $\\gamma\\in\\mathbb{C}\\setminus\\mathbb{R}$ and $|\\gamma|>1$, we show that for any non-real $\\gamma$ and $m < |\\gamma|^2-1$, the set $X^m(\\gamma)$ is not relatively dense in the complex plane.\n  Then we focus on complex Pisot units with a positive real conjugate"},"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":"1312.0653","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-12-02T22:55:14Z","cross_cats_sorted":["cs.DM","math.CO","math.NT"],"title_canon_sha256":"a73813836c045c10d807bb765381c6fe6220ef4dabf69dd926afc42aef72a947","abstract_canon_sha256":"d764964ae6ae380bdc3e667914462b461ef407c945ed3dd8d15e9d8ec9b6bcf5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:23.059760Z","signature_b64":"7zuODXmul+xmFUvCmZa4UNL36tvlg+1uLBLg4fKMxR1OYUW2/KikfRKUwIJ8tEkc20ivplhqYnO0f5XkJzGSCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89b4731e9779a973a19a0d05542f8a6dd5901c729ad11628ee9e760b13a6d427","last_reissued_at":"2026-05-18T02:44:23.059351Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:23.059351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral properties of cubic complex Pisot units","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO","math.NT"],"primary_cat":"math.MG","authors_text":"Edita Pelantov\\'a, Tom\\'a\\v{s} Hejda","submitted_at":"2013-12-02T22:55:14Z","abstract_excerpt":"For a real number $\\beta>1$, Erd\\H{o}s, Jo\\'o and Komornik study distances between consecutive points in the set $X^m(\\beta)=\\bigl\\{\\sum_{j=0}^n a_j \\beta^j : n\\in\\mathbb N,\\,a_j\\in\\{0,1,\\dots,m\\}\\bigr\\}$. Pisot numbers play a crucial role for the properties of $X^m(\\beta)$. Following the work of Za\\\"imi, who considered $X^m(\\gamma)$ with $\\gamma\\in\\mathbb{C}\\setminus\\mathbb{R}$ and $|\\gamma|>1$, we show that for any non-real $\\gamma$ and $m < |\\gamma|^2-1$, the set $X^m(\\gamma)$ is not relatively dense in the complex plane.\n  Then we focus on complex Pisot units with a positive real conjugate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0653","kind":"arxiv","version":3},"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":"1312.0653","created_at":"2026-05-18T02:44:23.059414+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.0653v3","created_at":"2026-05-18T02:44:23.059414+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.0653","created_at":"2026-05-18T02:44:23.059414+00:00"},{"alias_kind":"pith_short_12","alias_value":"RG2HGHUXPGUX","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_16","alias_value":"RG2HGHUXPGUXHIM2","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_8","alias_value":"RG2HGHUX","created_at":"2026-05-18T12:27:57.521954+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/RG2HGHUXPGUXHIM2BUCVIL4KNX","json":"https://pith.science/pith/RG2HGHUXPGUXHIM2BUCVIL4KNX.json","graph_json":"https://pith.science/api/pith-number/RG2HGHUXPGUXHIM2BUCVIL4KNX/graph.json","events_json":"https://pith.science/api/pith-number/RG2HGHUXPGUXHIM2BUCVIL4KNX/events.json","paper":"https://pith.science/paper/RG2HGHUX"},"agent_actions":{"view_html":"https://pith.science/pith/RG2HGHUXPGUXHIM2BUCVIL4KNX","download_json":"https://pith.science/pith/RG2HGHUXPGUXHIM2BUCVIL4KNX.json","view_paper":"https://pith.science/paper/RG2HGHUX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.0653&json=true","fetch_graph":"https://pith.science/api/pith-number/RG2HGHUXPGUXHIM2BUCVIL4KNX/graph.json","fetch_events":"https://pith.science/api/pith-number/RG2HGHUXPGUXHIM2BUCVIL4KNX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RG2HGHUXPGUXHIM2BUCVIL4KNX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RG2HGHUXPGUXHIM2BUCVIL4KNX/action/storage_attestation","attest_author":"https://pith.science/pith/RG2HGHUXPGUXHIM2BUCVIL4KNX/action/author_attestation","sign_citation":"https://pith.science/pith/RG2HGHUXPGUXHIM2BUCVIL4KNX/action/citation_signature","submit_replication":"https://pith.science/pith/RG2HGHUXPGUXHIM2BUCVIL4KNX/action/replication_record"}},"created_at":"2026-05-18T02:44:23.059414+00:00","updated_at":"2026-05-18T02:44:23.059414+00:00"}