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It is known that $E(x)$ is one of the following types of sets:\n  * finite union of closed intervals,\n  * homeomorphic to the Cantor set,\n  * homeomorphic to the set $T$ of subsums of $\\sum_{n=1}^{\\infty} c(n)$ where $c(2n-1)=\\frac{3}{4^n}$ and $c(2n)=\\frac{2}{4^n}$ (Cantorval).\n  Based on ideas of Jones and Velleman, and Guthrie and Nymann we describe families of sequences which contain, according to our knowledge, all known examples of $x$'s with $E(x)$"},"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":"1304.4218","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-11T06:30:09Z","cross_cats_sorted":[],"title_canon_sha256":"8a8ba32a8ecc644642f7bcb66c7a560163541cba5cfe14398d484fc9e0ffb9ac","abstract_canon_sha256":"66bfd7bdfcce7b6c3bcea27f734511fb4e31e039bf46611d1c9f2791f0806194"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:32.937845Z","signature_b64":"lFNtSUbKLhzxsYushtI1YFqSa5uEnA0kG77KzxWACREyNtTwxjWxmwu9baO2YNDMNFAEPuJDyK4ruGCNtZarBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb5494c820b2498097de7146c449071afe969a0c00c296989e4492b3d2d1ea23","last_reissued_at":"2026-05-18T01:09:32.937347Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:32.937347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multigeometric sequences and Cantorvals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Artur Bartoszewicz, Emilia Szymonik, Ma{\\l}gorzata Filipczak","submitted_at":"2013-04-11T06:30:09Z","abstract_excerpt":"For a sequence $x \\in l_1 \\setminus c_{00}$, one can consider the achievement set $E(x)$ of all subsums of series $\\sum_{n=1}^{\\infty} x(n)$. It is known that $E(x)$ is one of the following types of sets:\n  * finite union of closed intervals,\n  * homeomorphic to the Cantor set,\n  * homeomorphic to the set $T$ of subsums of $\\sum_{n=1}^{\\infty} c(n)$ where $c(2n-1)=\\frac{3}{4^n}$ and $c(2n)=\\frac{2}{4^n}$ (Cantorval).\n  Based on ideas of Jones and Velleman, and Guthrie and Nymann we describe families of sequences which contain, according to our knowledge, all known examples of $x$'s with $E(x)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4218","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":"1304.4218","created_at":"2026-05-18T01:09:32.937438+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.4218v2","created_at":"2026-05-18T01:09:32.937438+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4218","created_at":"2026-05-18T01:09:32.937438+00:00"},{"alias_kind":"pith_short_12","alias_value":"XNKJJSBAWJEY","created_at":"2026-05-18T12:28:06.772260+00:00"},{"alias_kind":"pith_short_16","alias_value":"XNKJJSBAWJEYBF66","created_at":"2026-05-18T12:28:06.772260+00:00"},{"alias_kind":"pith_short_8","alias_value":"XNKJJSBA","created_at":"2026-05-18T12:28:06.772260+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/XNKJJSBAWJEYBF66OFDMISIHDL","json":"https://pith.science/pith/XNKJJSBAWJEYBF66OFDMISIHDL.json","graph_json":"https://pith.science/api/pith-number/XNKJJSBAWJEYBF66OFDMISIHDL/graph.json","events_json":"https://pith.science/api/pith-number/XNKJJSBAWJEYBF66OFDMISIHDL/events.json","paper":"https://pith.science/paper/XNKJJSBA"},"agent_actions":{"view_html":"https://pith.science/pith/XNKJJSBAWJEYBF66OFDMISIHDL","download_json":"https://pith.science/pith/XNKJJSBAWJEYBF66OFDMISIHDL.json","view_paper":"https://pith.science/paper/XNKJJSBA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.4218&json=true","fetch_graph":"https://pith.science/api/pith-number/XNKJJSBAWJEYBF66OFDMISIHDL/graph.json","fetch_events":"https://pith.science/api/pith-number/XNKJJSBAWJEYBF66OFDMISIHDL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XNKJJSBAWJEYBF66OFDMISIHDL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XNKJJSBAWJEYBF66OFDMISIHDL/action/storage_attestation","attest_author":"https://pith.science/pith/XNKJJSBAWJEYBF66OFDMISIHDL/action/author_attestation","sign_citation":"https://pith.science/pith/XNKJJSBAWJEYBF66OFDMISIHDL/action/citation_signature","submit_replication":"https://pith.science/pith/XNKJJSBAWJEYBF66OFDMISIHDL/action/replication_record"}},"created_at":"2026-05-18T01:09:32.937438+00:00","updated_at":"2026-05-18T01:09:32.937438+00:00"}