{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:PH6WG7YAPYTYL2DZLCPHA2ZST4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"60d838a460185706b89ed789bb6bd80dd5daa9e0cfd48a4782d3ed5896b0e0af","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-20T08:21:57Z","title_canon_sha256":"1128901d4d91b1e8f5ec6316e2c0615f4a420850bddc3fe37319b7f1f31ea28f"},"schema_version":"1.0","source":{"id":"1905.07938","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.07938","created_at":"2026-05-17T23:45:48Z"},{"alias_kind":"arxiv_version","alias_value":"1905.07938v1","created_at":"2026-05-17T23:45:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.07938","created_at":"2026-05-17T23:45:48Z"},{"alias_kind":"pith_short_12","alias_value":"PH6WG7YAPYTY","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"PH6WG7YAPYTYL2DZ","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"PH6WG7YA","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:56059e0082a5afa0a05e1eef1cabdb0db380ca4cc44d4897352b2da8d2d3664d","target":"graph","created_at":"2026-05-17T23:45:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\\leq\\alpha\\leq\\beta\\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that $\\mathrm{d}(A)=\\alpha$ and $\\mathrm{d}(A+A)=\\beta$. More generally we study the set of $k$-tuples $(\\mathrm{d}(iA))_{1\\leq i\\leq k}$ for $A\\subset \\mathbb{N}$. This leads us to introduce subsets defined by diophantine constraints inside a random set of integers known as the set of ``pseudo $s$th powers''. We consider similar problems for subsets of the circle $\\mathb","authors_text":"Fran\\c{c}ois Hennecart, Pierre-Yves Bienvenu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-20T08:21:57Z","title":"On the density or measure of sets and their sumsets in the integers or the circle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.07938","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:11bdc4ba47c6a4ced950c2451c64dad8b176e8ba0a61ae0f3e077fe98172d166","target":"record","created_at":"2026-05-17T23:45:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"60d838a460185706b89ed789bb6bd80dd5daa9e0cfd48a4782d3ed5896b0e0af","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-20T08:21:57Z","title_canon_sha256":"1128901d4d91b1e8f5ec6316e2c0615f4a420850bddc3fe37319b7f1f31ea28f"},"schema_version":"1.0","source":{"id":"1905.07938","kind":"arxiv","version":1}},"canonical_sha256":"79fd637f007e2785e879589e706b329f2fc6cfa1365bd999aeffbc73268478dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"79fd637f007e2785e879589e706b329f2fc6cfa1365bd999aeffbc73268478dd","first_computed_at":"2026-05-17T23:45:48.022111Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:45:48.022111Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LkjuVwl+LrsxHawFiNw8QrysuL9D6D+iX/VTPW6ijUUBswQrICjD1ZNmirP/KcXfS1Xhf8U3wNpsatCkWCaYAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:45:48.022637Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.07938","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:11bdc4ba47c6a4ced950c2451c64dad8b176e8ba0a61ae0f3e077fe98172d166","sha256:56059e0082a5afa0a05e1eef1cabdb0db380ca4cc44d4897352b2da8d2d3664d"],"state_sha256":"4ab78bb2b1e5deb04f60d10a4212cb7df130521a7238f0c12430396ff92e87e7"}