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For the dual numbers we expose a range where the minimum value of $\\max\\{|A+A|,|AA|\\}$ is neither close to $|A|$ nor to $|A|^2$.\n  To obtain our main sum-product bound, we extend Elekes' sum-product technique that relies on point-li"},"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":"1812.09547","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-22T15:45:15Z","cross_cats_sorted":[],"title_canon_sha256":"293ec5d4ffa3f68c16701dcaebd1bd260764b4569881da53555b2aff948c8c46","abstract_canon_sha256":"94cdef86056dbd91986f6f8ed6d870e13b6887082f261349ac8e2b465b7df734"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:27.584549Z","signature_b64":"hzdEKN47P+Qx20l6u8T6Fdxgqt8bF+3ytPJ5lEgxpOquiuFA889aIOJk7aL5vMToJHpwdp/XaBlLOc6Y1sXBCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d3e6a04a9a0b2cb0097d075ef1307ea177abb0d3f4073c429630331f5366259","last_reissued_at":"2026-05-17T23:57:27.583882Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:27.583882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sum-Product Phenomena for Planar Hypercomplex Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Matthew Hase-Liu","submitted_at":"2018-12-22T15:45:15Z","abstract_excerpt":"We study the sum-product problem for the planar hypercomplex numbers: the dual numbers and double numbers. 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