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In 2012, the first author, Angel, the second author, and Lubetzky proved that, for expander graphs and $H=\\mathbb{Z}$, this minimum is at least $\\Omega(\\log n)$, and this bound is tight -- there exists a regular expander $G$ with ${\\sf S}_{\\mathbb{Z}}(G)=O(\\log n)$. We prove that, for every constant $d\\geq 3$, the random $d$-regular graph $\\mathcal{G}_{n,d}$ has significantly larger sum-sets: with high probability, for"},"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":"2507.01138","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-07-01T18:56:07Z","cross_cats_sorted":[],"title_canon_sha256":"ac148726b482571765c875b3b74473bbbb83bf06246fb6693161388947b4e833","abstract_canon_sha256":"3c56c3aa49f5068004efe2c4364b4f48c51be18c38adbb77ce5e4f7663119ff0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:46:44.713246Z","signature_b64":"enkcedft1YClYxPitUnuNZ6+hfq2k7xMN3zdnYESPHAC2PQbQpefD79b/q0rMnKJO8g0q5eivnC1NZVmvD70CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f32087ead2b458635a80b3b9530fc966bf5bce979c1bf7fe0b54861dfdd5ad21","last_reissued_at":"2026-07-05T11:46:44.712642Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:46:44.712642Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums along the edges of bounded degree graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Georgii Zakharov, Itai Benjamini, Maksim Zhukovskii, Noga Alon","submitted_at":"2025-07-01T18:56:07Z","abstract_excerpt":"Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. 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We prove that, for every constant $d\\geq 3$, the random $d$-regular graph $\\mathcal{G}_{n,d}$ has significantly larger sum-sets: with high probability, for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.01138","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.01138/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2507.01138","created_at":"2026-07-05T11:46:44.712720+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.01138v2","created_at":"2026-07-05T11:46:44.712720+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.01138","created_at":"2026-07-05T11:46:44.712720+00:00"},{"alias_kind":"pith_short_12","alias_value":"6MQIP2WSWRMG","created_at":"2026-07-05T11:46:44.712720+00:00"},{"alias_kind":"pith_short_16","alias_value":"6MQIP2WSWRMGGWUA","created_at":"2026-07-05T11:46:44.712720+00:00"},{"alias_kind":"pith_short_8","alias_value":"6MQIP2WS","created_at":"2026-07-05T11:46:44.712720+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.06358","citing_title":"Universality in random graphs via optimal linking systems: trees and beyond","ref_index":5,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2","json":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2.json","graph_json":"https://pith.science/api/pith-number/6MQIP2WSWRMGGWUAWO4VGD6JM2/graph.json","events_json":"https://pith.science/api/pith-number/6MQIP2WSWRMGGWUAWO4VGD6JM2/events.json","paper":"https://pith.science/paper/6MQIP2WS"},"agent_actions":{"view_html":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2","download_json":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2.json","view_paper":"https://pith.science/paper/6MQIP2WS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.01138&json=true","fetch_graph":"https://pith.science/api/pith-number/6MQIP2WSWRMGGWUAWO4VGD6JM2/graph.json","fetch_events":"https://pith.science/api/pith-number/6MQIP2WSWRMGGWUAWO4VGD6JM2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2/action/storage_attestation","attest_author":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2/action/author_attestation","sign_citation":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2/action/citation_signature","submit_replication":"https://pith.science/pith/6MQIP2WSWRMGGWUAWO4VGD6JM2/action/replication_record"}},"created_at":"2026-07-05T11:46:44.712720+00:00","updated_at":"2026-07-05T11:46:44.712720+00:00"}