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In addition, we extend a result of Shkredov to show that $|2A|\\gg |A|^{\\frac 85-\\epsilon}$ for $|A|\\ll p^{\\frac 59}$."},"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":"1303.2729","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-12T00:15:39Z","cross_cats_sorted":[],"title_canon_sha256":"fa2ca413e626b2f67e7aceb22d8835e3013090ac096f7e4a7244b7ba65207de1","abstract_canon_sha256":"924c5ffafe18d7689a0b1bf5487a28f10026f5964b5ef21f999d6ec79e19b8ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:05.573357Z","signature_b64":"9H0AQwuV500dwjZBjutvkig3OkAN87VcrfVZPWWQtUvxejOD0RnkMxRwNqcgyT05pTdb6y4QTjcIln+kHYuFBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e146fdc87aa15e5383d501b80b22a1c4811dc5a0d852528504edcf0099d92de","last_reissued_at":"2026-05-18T03:25:05.572616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:05.572616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on sumsets of subgroups in $\\mathbb Z_p^*$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Derrick Hart","submitted_at":"2013-03-12T00:15:39Z","abstract_excerpt":"Let $A$ be a multiplicative subgroup of $\\mathbb Z_p^*$. 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