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Guthrie and Nymann proved that $E(x)$ is one of the following types of sets:\n  (I) a finite union of closed intervals;\n  (C) homeomorphic to the Cantor set;\n  (MC) homeomorphic to the set $T$ of subsums of $\\sum_{n=1}^\\infty b(n)$ where $b(2n-1) = 3/4^n$ and $b(2n) = 2/4^n$.\n  By $I$, $C$ and $MC$ we denote the sets of all sequences $x \\in l_1 \\setminus c_{00}$, such that $E(x)$ has the corresponding property. In this note we show that $I$ and $C$ are strongly $\\mat"},"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":"1208.3058","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-08-15T08:30:49Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"d3b2ae98463dfb8764033d95847953e969309e5dbef6875102fa1b2be16d8dad","abstract_canon_sha256":"91d3af4212dae149bd647643a42741572c8cfcd236588ee2bac3abc60979c3bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:24:54.130495Z","signature_b64":"Go3nImvCzCQgKBtwzmnDgVR/LT3wWsTjthEccDsZ1UjtffZ8sW13cp4ZSt8c8g2ggfNLP0i5zIzIKT6xb3ABDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1824a1b0003fcb49c8f8e88cd09a46766af9ed5916aed90cdb98db3f64870e0d","last_reissued_at":"2026-05-18T03:24:54.129991Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:24:54.129991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebraic and topological properties of some sets in $l_1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.GN","authors_text":"A. Bartoszewicz, E. Szymonik, Sz. Glab, T. Banakh","submitted_at":"2012-08-15T08:30:49Z","abstract_excerpt":"For a sequence $x \\in l_1 \\setminus c_{00}$, one can consider the set $E(x)$ of all subsums of series $\\sum_{n=1}^{\\infty} x(n)$. Guthrie and Nymann proved that $E(x)$ is one of the following types of sets:\n  (I) a finite union of closed intervals;\n  (C) homeomorphic to the Cantor set;\n  (MC) homeomorphic to the set $T$ of subsums of $\\sum_{n=1}^\\infty b(n)$ where $b(2n-1) = 3/4^n$ and $b(2n) = 2/4^n$.\n  By $I$, $C$ and $MC$ we denote the sets of all sequences $x \\in l_1 \\setminus c_{00}$, such that $E(x)$ has the corresponding property. 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