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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","authors_text":"Jiali Zhu, Kan Jiang, Li Tian, Xiaomin Ren","cross_cats":["math.MG","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-11T09:43:06Z","title":"Multiple representations of real numbers on self-similar sets with overlaps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04930","kind":"arxiv","version":2},"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:eb3846a4fa569ea7858ec53a38669b948f4c4a5c3a33791f5a69f6a1bbf4edae","target":"record","created_at":"2026-05-17T23:56:05Z","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":"465dc625ae99108d2a6dd24c760c0ee4f4427a5ba6f0257868f869c9558b03af","cross_cats_sorted":["math.MG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-11T09:43:06Z","title_canon_sha256":"e45dda0ab11256864c93e5ceeea23c634a54cf751989af208c5ca8e0f633daab"},"schema_version":"1.0","source":{"id":"1810.04930","kind":"arxiv","version":2}},"canonical_sha256":"6d7c02a3d594ce46517c87819a57006e4d4ca9a79997be51727de263d862b3f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d7c02a3d594ce46517c87819a57006e4d4ca9a79997be51727de263d862b3f1","first_computed_at":"2026-05-17T23:56:05.141623Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:05.141623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2NUaPhNwSVsyw6ZI60lpO7RdXH15T5OFiMqJisEJStOhTdUB/523XpfN/CTroBcXNDyY9yNK9RHJZ5qNgn0HCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:05.142136Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.04930","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eb3846a4fa569ea7858ec53a38669b948f4c4a5c3a33791f5a69f6a1bbf4edae","sha256:2ae4a8548533be29d15f31c35320ccf1c3a51af4481165fc1d58810458f62cd8"],"state_sha256":"d749ed1cd14367f25221357a67c749447a8500f2f9116ff904526928f894699a"}