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Let $A/F$ be an abelian variety, $f:A \\rightarrow \\mathbb{P}^n$ be a morphism which is finite onto its image, and $\\Gamma \\subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\\mathbb{A}^n \\subseteq \\mathbb{P}^n$ and any finite subset $X \\subseteq f(\\Gamma) \\cap \\mathbb{A}^n$, the energy satisfies $E(X) \\ll \\lvert X \\rvert^2$ and the sumset satisfies $\\lvert X+X \\rvert \\gg \\lvert X \\rvert^2$. Thus images of finite-r"},"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":"2606.01299","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-31T15:39:57Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"0755ad3bb639e231296732856e2b7e890841507c04b618c5f402d352a4b48c95","abstract_canon_sha256":"b439dca6cc3436f3b5c7ab1d19a269e97242d1999a590860f9afface8653f5fb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:29.441205Z","signature_b64":"qmzCc4vwNE7G8KxypS8VEXB2tg3vvAVo/Q1SgQ3CPNJ4dDcdmnO4G+fKcPtSnP3WHUfHTuHwOEfr/jjCedk8DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95a8d19e386b4d57c35bd0362620fd5cb586f266f6730f7bc8269aded805bccf","last_reissued_at":"2026-06-02T02:04:29.440178Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:29.440178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Additive Rigidity for Images of Rational Points on Abelian Varieties II: The General Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Seokhyun Choi","submitted_at":"2026-05-31T15:39:57Z","abstract_excerpt":"We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to a projective space. Let $A/F$ be an abelian variety, $f:A \\rightarrow \\mathbb{P}^n$ be a morphism which is finite onto its image, and $\\Gamma \\subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\\mathbb{A}^n \\subseteq \\mathbb{P}^n$ and any finite subset $X \\subseteq f(\\Gamma) \\cap \\mathbb{A}^n$, the energy satisfies $E(X) \\ll \\lvert X \\rvert^2$ and the sumset satisfies $\\lvert X+X \\rvert \\gg \\lvert X \\rvert^2$. 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