{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4KZ2U6WNOC7GTE3UW2L3ZPC3JF","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":"f3b77652b0e3bab57915c0c3e35f7274dc5e380f65d9eeb073dd51517c597445","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-22T22:46:58Z","title_canon_sha256":"32c1fb50937c917d439a97f4d9ebc7e141e40c793324839d0e746d192b8c616a"},"schema_version":"1.0","source":{"id":"1312.6438","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.6438","created_at":"2026-05-18T03:02:58Z"},{"alias_kind":"arxiv_version","alias_value":"1312.6438v2","created_at":"2026-05-18T03:02:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.6438","created_at":"2026-05-18T03:02:58Z"},{"alias_kind":"pith_short_12","alias_value":"4KZ2U6WNOC7G","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4KZ2U6WNOC7GTE3U","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4KZ2U6WN","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:2ac2f806ec41c1dceb5b4ea42a159a83caf0676703aba3362b25392d76aa59a3","target":"graph","created_at":"2026-05-18T03:02:58Z","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":"This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\\subset{\\mathbb{R}}$, $$|A(A+A)|\\gg{|A|^{\\frac{3}{2}+\\frac{1}{178}}},$$ giving a partial answer to a conjecture of Balog. In a similar spirit, it is established that $$|A(A+A+A+A)|\\gg{\\frac{|A|^2}{\\log{|A|}}},$$ a bound which is optimal up to constant and logarithmic factors. We also prove several new results concerning sum-product estimates and expanders, for example, showing that $$|A(A+a)|\\gg{|A|^{3/2}}$$ holds for a typical element of $A$.","authors_text":"Brendan Murphy, Ilya D. Shkredov, Oliver Roche-Newton","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-22T22:46:58Z","title":"Variations on the Sum-Product Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6438","kind":"arxiv","version":2},"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:ce484892fcc4a99a4f510452401e17f3ac72513ecae8a7f10082b14cc1f356f6","target":"record","created_at":"2026-05-18T03:02:58Z","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":"f3b77652b0e3bab57915c0c3e35f7274dc5e380f65d9eeb073dd51517c597445","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-22T22:46:58Z","title_canon_sha256":"32c1fb50937c917d439a97f4d9ebc7e141e40c793324839d0e746d192b8c616a"},"schema_version":"1.0","source":{"id":"1312.6438","kind":"arxiv","version":2}},"canonical_sha256":"e2b3aa7acd70be699374b697bcbc5b495ddf99f3b611d3642c84ef2534b6363c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e2b3aa7acd70be699374b697bcbc5b495ddf99f3b611d3642c84ef2534b6363c","first_computed_at":"2026-05-18T03:02:58.847975Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:58.847975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O+0c5X/1y945NWMB4R07/+689OG7wamr/zoZy4MYma+LmxpviSnVqY6KhXikWWIJiNzSojtrxskpl6xvLyj8Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:58.848865Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.6438","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce484892fcc4a99a4f510452401e17f3ac72513ecae8a7f10082b14cc1f356f6","sha256:2ac2f806ec41c1dceb5b4ea42a159a83caf0676703aba3362b25392d76aa59a3"],"state_sha256":"c0b74537d5eb1e6be7ff87b31af052b01bcba2f6a9e75b4db2efafc7a5cae4f5"}