{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:NV6AFI6VSTHEMUL4Q6AZUVYANZ","short_pith_number":"pith:NV6AFI6V","schema_version":"1.0","canonical_sha256":"6d7c02a3d594ce46517c87819a57006e4d4ca9a79997be51727de263d862b3f1","source":{"kind":"arxiv","id":"1810.04930","version":2},"attestation_state":"computed","paper":{"title":"Multiple representations of real numbers on self-similar sets with overlaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Jiali Zhu, Kan Jiang, Li Tian, Xiaomin Ren","submitted_at":"2018-10-11T09:43:06Z","abstract_excerpt":"Let $K$ be the attractor of the following IFS\n  $$\\{f_1(x)=\\lambda x, f_2(x)=\\lambda x +c-\\lambda,f_3(x)=\\lambda x +1-\\lambda\\}, $$\n  where $f_1(I)\\cap f_2(I)\\neq \\emptyset, (f_1(I)\\cup f_2(I))\\cap f_3(I)=\\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main results of this paper are as follows: $$\\sqrt{K}+\\sqrt{K}=[0,2]$$ if and only if $$\\sqrt{c}+1\\geq 2\\sqrt{1-\\lambda},$$ where $\\sqrt{K}+\\sqrt{K}=\\{\\sqrt{x}+\\sqrt{y}:x,y\\in K\\}$. If $c\\geq (1-\\lambda)^2$, then $$\\dfrac{K}{K}=\\left\\{\\dfrac{x}{y}:x,y\\in K, y\\neq 0\\right\\}=\\left[0,\\infty\\right).$$ As a consequence, we prove that the fol"},"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":"1810.04930","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-11T09:43:06Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"e45dda0ab11256864c93e5ceeea23c634a54cf751989af208c5ca8e0f633daab","abstract_canon_sha256":"465dc625ae99108d2a6dd24c760c0ee4f4427a5ba6f0257868f869c9558b03af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:05.142136Z","signature_b64":"2NUaPhNwSVsyw6ZI60lpO7RdXH15T5OFiMqJisEJStOhTdUB/523XpfN/CTroBcXNDyY9yNK9RHJZ5qNgn0HCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d7c02a3d594ce46517c87819a57006e4d4ca9a79997be51727de263d862b3f1","last_reissued_at":"2026-05-17T23:56:05.141623Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:05.141623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple representations of real numbers on self-similar sets with overlaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Jiali Zhu, Kan Jiang, Li Tian, Xiaomin Ren","submitted_at":"2018-10-11T09:43:06Z","abstract_excerpt":"Let $K$ be the attractor of the following IFS\n  $$\\{f_1(x)=\\lambda x, f_2(x)=\\lambda x +c-\\lambda,f_3(x)=\\lambda x +1-\\lambda\\}, $$\n  where $f_1(I)\\cap f_2(I)\\neq \\emptyset, (f_1(I)\\cup f_2(I))\\cap f_3(I)=\\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main results of this paper are as follows: $$\\sqrt{K}+\\sqrt{K}=[0,2]$$ if and only if $$\\sqrt{c}+1\\geq 2\\sqrt{1-\\lambda},$$ where $\\sqrt{K}+\\sqrt{K}=\\{\\sqrt{x}+\\sqrt{y}:x,y\\in K\\}$. If $c\\geq (1-\\lambda)^2$, then $$\\dfrac{K}{K}=\\left\\{\\dfrac{x}{y}:x,y\\in K, y\\neq 0\\right\\}=\\left[0,\\infty\\right).$$ As a consequence, we prove that the fol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04930","kind":"arxiv","version":2},"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":"1810.04930","created_at":"2026-05-17T23:56:05.141704+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.04930v2","created_at":"2026-05-17T23:56:05.141704+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.04930","created_at":"2026-05-17T23:56:05.141704+00:00"},{"alias_kind":"pith_short_12","alias_value":"NV6AFI6VSTHE","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"NV6AFI6VSTHEMUL4","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"NV6AFI6V","created_at":"2026-05-18T12:32:40.477152+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/NV6AFI6VSTHEMUL4Q6AZUVYANZ","json":"https://pith.science/pith/NV6AFI6VSTHEMUL4Q6AZUVYANZ.json","graph_json":"https://pith.science/api/pith-number/NV6AFI6VSTHEMUL4Q6AZUVYANZ/graph.json","events_json":"https://pith.science/api/pith-number/NV6AFI6VSTHEMUL4Q6AZUVYANZ/events.json","paper":"https://pith.science/paper/NV6AFI6V"},"agent_actions":{"view_html":"https://pith.science/pith/NV6AFI6VSTHEMUL4Q6AZUVYANZ","download_json":"https://pith.science/pith/NV6AFI6VSTHEMUL4Q6AZUVYANZ.json","view_paper":"https://pith.science/paper/NV6AFI6V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.04930&json=true","fetch_graph":"https://pith.science/api/pith-number/NV6AFI6VSTHEMUL4Q6AZUVYANZ/graph.json","fetch_events":"https://pith.science/api/pith-number/NV6AFI6VSTHEMUL4Q6AZUVYANZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NV6AFI6VSTHEMUL4Q6AZUVYANZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NV6AFI6VSTHEMUL4Q6AZUVYANZ/action/storage_attestation","attest_author":"https://pith.science/pith/NV6AFI6VSTHEMUL4Q6AZUVYANZ/action/author_attestation","sign_citation":"https://pith.science/pith/NV6AFI6VSTHEMUL4Q6AZUVYANZ/action/citation_signature","submit_replication":"https://pith.science/pith/NV6AFI6VSTHEMUL4Q6AZUVYANZ/action/replication_record"}},"created_at":"2026-05-17T23:56:05.141704+00:00","updated_at":"2026-05-17T23:56:05.141704+00:00"}