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For a nonempty finite subset $A=\\{a_0, a_1,\\ldots, a_{k-1}\\}$ of $G$, we let \\[h_{\\underline{+}}A:=\\{\\Sigma_{i=0}^{k-1}\\lambda_{i} a_{i}: (\\lambda_{0}, \\ldots, \\lambda_{k-1}) \\in \\mathbb{Z}^{k},~ \\Sigma_{i=0}^{k-1}|\\lambda_{i}|=h \\},\\] be the {\\it signed sumset} of $A$.\n  The {\\it direct problem} for the signed sumset $h_{\\underline{+}}A$ is to find a nontrivial lower bound for $|h_{\\underline{+}}A|$ in terms of $|A|$. The {\\it inverse problem} for $h_{\\underline{+}}A$ is to determine the structure of the finite set $A$ for wh"},"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":"1810.02673","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-05T13:32:12Z","cross_cats_sorted":[],"title_canon_sha256":"7ef873584b6e99e7c2533c1749f5fd8a04f09927f67bf22a584ac76cad2aa66d","abstract_canon_sha256":"668c92f0368809887965e968e0800d399b8d345c8dc8d648628061e3bfe61ee4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:01.047153Z","signature_b64":"84Z5noIrThwRsu1O97JYfDTQt5ivw1jGXm3zADvKJ44ds7dZjVmUxrYDHUbVZRvzSLt7BwCy/vRy35CUM4s9CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00c31f3e7f868866978a41a68240a5a6f69086defc911781ef530a00d85b526a","last_reissued_at":"2026-05-18T00:04:01.046608Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:01.046608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Direct and Inverse Theorems on Signed Sumsets of Integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jagannath Bhanja, Ram Krishna Pandey","submitted_at":"2018-10-05T13:32:12Z","abstract_excerpt":"Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\\{a_0, a_1,\\ldots, a_{k-1}\\}$ of $G$, we let \\[h_{\\underline{+}}A:=\\{\\Sigma_{i=0}^{k-1}\\lambda_{i} a_{i}: (\\lambda_{0}, \\ldots, \\lambda_{k-1}) \\in \\mathbb{Z}^{k},~ \\Sigma_{i=0}^{k-1}|\\lambda_{i}|=h \\},\\] be the {\\it signed sumset} of $A$.\n  The {\\it direct problem} for the signed sumset $h_{\\underline{+}}A$ is to find a nontrivial lower bound for $|h_{\\underline{+}}A|$ in terms of $|A|$. 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