{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:24QIY3WONMCZSFHYUWG5EGWA6K","short_pith_number":"pith:24QIY3WO","schema_version":"1.0","canonical_sha256":"d7208c6ece6b059914f8a58dd21ac0f2825bad66640c04227d78f8f9e68bc687","source":{"kind":"arxiv","id":"2605.28973","version":1},"attestation_state":"computed","paper":{"title":"Asymptotic formulas for sums of elements from a multiplicative group","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, Jan-Hendrik Evertse, K\\'alm\\'an Gy\\H{o}ry, Lajos Hajdu, L\\'aszl\\'o Remete","submitted_at":"2026-05-27T18:22:59Z","abstract_excerpt":"Let $K$ be a number field, $k\\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $\\Gamma$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(\\alpha )$ denote the absolute exponential height of an algebraic number $\\alpha$. Fix non-zero elements $a_1\\kdots a_k\\in K$. We give asymptotic formulas for the number of $\\mathbf{x}=(x_1\\kdots x_k)\\in\\Gamma$ with $H(a_1x_1+\\cdots +a_kx_k)\\leq X$ as $X\\to\\infty$ such that no non-empty subsum of $a_1x_1+\\cdots +a_kx_k$ vanishes. By the same method of proof, we obtain an asymptotic"},"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":"2605.28973","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-27T18:22:59Z","cross_cats_sorted":[],"title_canon_sha256":"62e5c689219b2af288a3383e7922adda7723eb2c634cd9bba9c28fa62147b172","abstract_canon_sha256":"3b0cb631ed71495ceb78e2899e812c642165b559c042c2e507a172515ac930e6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T01:04:42.101816Z","signature_b64":"Jdk+7A1FBTVok2D5CECqZ9PGfJPJCmgo/qpUKbEPYOTElznW+LFnGz+cxNWLPwGwrCFbFVr5v/dzEIdGN7oWBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7208c6ece6b059914f8a58dd21ac0f2825bad66640c04227d78f8f9e68bc687","last_reissued_at":"2026-05-29T01:04:42.101402Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T01:04:42.101402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic formulas for sums of elements from a multiplicative group","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, Jan-Hendrik Evertse, K\\'alm\\'an Gy\\H{o}ry, Lajos Hajdu, L\\'aszl\\'o Remete","submitted_at":"2026-05-27T18:22:59Z","abstract_excerpt":"Let $K$ be a number field, $k\\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $\\Gamma$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(\\alpha )$ denote the absolute exponential height of an algebraic number $\\alpha$. Fix non-zero elements $a_1\\kdots a_k\\in K$. We give asymptotic formulas for the number of $\\mathbf{x}=(x_1\\kdots x_k)\\in\\Gamma$ with $H(a_1x_1+\\cdots +a_kx_k)\\leq X$ as $X\\to\\infty$ such that no non-empty subsum of $a_1x_1+\\cdots +a_kx_k$ vanishes. By the same method of proof, we obtain an asymptotic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28973","kind":"arxiv","version":1},"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/2605.28973/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":"2605.28973","created_at":"2026-05-29T01:04:42.101466+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.28973v1","created_at":"2026-05-29T01:04:42.101466+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.28973","created_at":"2026-05-29T01:04:42.101466+00:00"},{"alias_kind":"pith_short_12","alias_value":"24QIY3WONMCZ","created_at":"2026-05-29T01:04:42.101466+00:00"},{"alias_kind":"pith_short_16","alias_value":"24QIY3WONMCZSFHY","created_at":"2026-05-29T01:04:42.101466+00:00"},{"alias_kind":"pith_short_8","alias_value":"24QIY3WO","created_at":"2026-05-29T01:04:42.101466+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K","json":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K.json","graph_json":"https://pith.science/api/pith-number/24QIY3WONMCZSFHYUWG5EGWA6K/graph.json","events_json":"https://pith.science/api/pith-number/24QIY3WONMCZSFHYUWG5EGWA6K/events.json","paper":"https://pith.science/paper/24QIY3WO"},"agent_actions":{"view_html":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K","download_json":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K.json","view_paper":"https://pith.science/paper/24QIY3WO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.28973&json=true","fetch_graph":"https://pith.science/api/pith-number/24QIY3WONMCZSFHYUWG5EGWA6K/graph.json","fetch_events":"https://pith.science/api/pith-number/24QIY3WONMCZSFHYUWG5EGWA6K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K/action/storage_attestation","attest_author":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K/action/author_attestation","sign_citation":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K/action/citation_signature","submit_replication":"https://pith.science/pith/24QIY3WONMCZSFHYUWG5EGWA6K/action/replication_record"}},"created_at":"2026-05-29T01:04:42.101466+00:00","updated_at":"2026-05-29T01:04:42.101466+00:00"}